Greta Garbo ate there once, apparently. Tina Turner has a tendency to embrace the waiters. Lucien Freud is hardly ever out of the place. Martina Navratilova might as well have had a season ticket, dining there whenever she won Wimbledon. Will Self goes a lot, without having to worry about anyone examining his pupils after a trip to the gents. John Mortimer and Jeremy Paxman and the rest of the Henley-on-Thames mafia sometimes take over the whole place for fund-raising quiz evenings. Mick Jagger and David Bowie have sat quietly examining the menu, like ordinary human beings, trying to endure the creak of turning heads. And if Tony Blair makes it to Downing Street on Friday, it won't be long before he too makes the journey west, to discuss the politicisation of the Mayorship of London with his friend Richard (Lord) Rogers.

It is, of course, the River Cafe - that cynosure of gastro-erotic Nineties London chic, that glossy, metallic Hammersmith home of cucina rustica, that is, Italian peasant food that no Italian peasant from Turin to Palermo could ever afford. This autumn, it celebrates 10 years of polenta and porcini, 10 years of unfeasibly thick, take-your-stomach-home-in-a-wheelbarrow bean soups and tidal waves of balsamic vinegar, 10 years of chargrilled this and drizzled that, of insalata and inzimino, of bruschetta and ricotta, pangrattato, cannellini...

And the women who own and run the River Cafe have become as famous as the place - though in a way that's significantly different from the standard- issue restaurateur. Ruth Rogers and Rose Gray do a lot of the cooking themselves, in starchy white uniforms with stray strands of sweaty hair flicked behind their ears; but they're known less as celebrity chefs hungry for Michelin rosettes than as joint hostesses of a busily arty salon. "The opening of our restaurant coincided with the decline of the pompous male chef who made you feel his restaurant was a temple of food, and everything had to be hushed and quiet," says Ruth Rogers, "That's so old-fashioned. Rose and I are both from large families, with lots of people sitting round the table and talking all the time. We wanted the place to be an extension of the way we ate with our families and friends. None of that `You're lucky to be here' attitude." "And from the start," said Rose, "we agreed that we wanted an American style of service - that smartness, those good looks, the combination of intelligence and love of the job..."

I went to have a look. The place was filling up fast by 12.45pm. Arriving lunchers had that look on their faces that says, "It's taken months to get a table, but I'm going to walk in as if I own the place." A famously ebullient Welsh writer was being lunched by her agent. Jonathan Newhouse, nephew of the God-like media mogul Si Newhouse (who owns the New Yorker) was meeting the proprietor of Viz and Bizarre magazines. While waiting for the co-owners, I marvelled at the names of the staff. The maitre d' is a dramatic Morticia Addams figure in all-over black, who answers to the name of "Limpet", a humorous contraction of Elizabeth. The hunk by the door is called Dante. He checks a booking in the ledger and summons a waitress of heartbreaking beauty with the words "Vashti - would you mind?" You suspect that if somebody called Jane ever got a job here, she would be magically renamed Fuschia or Nefertiti by nightfall.

Rose Gray appears. The older of the partners, she is lean-faced and headmistressy, but her conversation is warm and her eyes sparkle. She orders the staff around with cod-bossiness ("Out of the way, you lot. I saw you trying to sneak into the photograph..."). When the wine is poured for her inspection, she inserts her nose into the glass like a crane with a water-jug. Her expression changes. Her brows knit. Something is not right. You can almost hear the wine quaking with apprehension, nervy ripples on its oxidised surface. She packs the offending vintage off to Bottle Detention with utter confidence.

We are joined by Ruth Rogers. Looks-wise, she is frankly Gaby Roslin's elder sister, with straw-blonde hair and milky-blue eyes. A former Sixties radical and the wife of Richard Rogers, she is socially adroit and oceanically well-connected, from the Oval Office to the Foreign Office. They're an intriguing pair, Ruth a flatterer, Rose a straight-talker, both cautiously protective of the Cafe's reputation, both clearly delighted to have been so comprehensively "taken up" by the media establishment and the chef community alike.

"When we started," said Rose, "everyone said, `Oh, the restaurant business, it's so competitive.' But we grew into the generation of Kensington Place and Bibendum and the boys at Le Caprice and the Ivy, and they've all been very generous and supportive. If we have a problem, they'll help us out. All the great chefs of the world come through the River Cafe now - people like Michael Chow and Alice Waters [of Chez Panisse, the north Californian home of organic cooking and the River Cafe's biggest influence]. And even though we're not going in the same direction as Marco [Pierre White] and Nico [Ladenis] and creating food in complex ways, they recognise what we're trying to do." Which is? "Trying to cook Italian food to a sublime degree. Trying to do it to perfection."

"Perfectionism" is an odd word. It takes the concept of utter rightness and holds it at arm's length, as though there were something neurotic and pernickety about wanting such a thing. Ruth and Rose are perfectionists in ways that can seem both good and slightly absurd. About vegetables, for example, they are as doctrinaire as any tabloid-conceived EU commissar laying down the law about straight bananas. In the pages of their new book, River Cafe Cook Book Two, you learn, for instance, that beetroots must be "the size of golfballs", fig salad should comprise either "purple basil and ripe black figs or green basil and ripe green figs" but never a combination of the two. They even specify the kind of salt you should use (Maldon)... Did they believe in some Platonic theory of ideal food?

"Of course, there's such a thing as a perfect zucchini," said Rose Gray. "It has to be organically grown and picked when it tastes best, which is [she extended a bony forefinger] when it's slightly longer than your first finger, and before the seeds have developed inside. Because after they've developed, the flesh gets softer and watery and you won't get the intense flavour." "Our cooking is all about flavour," interjected Ruth. "If you have an anchovy that's salted or a zucchini that's marinaded, you're going to get the best out of it." I'd have thought it was a matter of taste, but only a madman would dream of contradicting Lady Rogers in full, idealistic flow. "What Rose and I want is for things to be in season, to cook them as close as possible to their best. We don't want raspberries in January, don't want microwaves, don't want anything frozen. And there's a certain excitement about saying to people, it's nearly May / June and the melons are coming soon. And then they'll be gone in a month's time and something else will be here. Right now the asparagus is at its best, and we'll cook it like mad for a while, they we'll stop and you won't get it for another year."

And in case you're wondering how the River Cafe ladies can sit in Hammersmith, London W6 and talk about their "zucchini" rather than their "courgettes", it's because of their scrupulousness about everything being genuinely Italian. Their vegetables may be grown by English suppliers, but they're grown from seeds acquired in Italy. "We look around," said Rose, "and go to the shops that sell the seeds that produced the vegetables that appear in the market. We're terribly selective: this particular pumpkin, these particular cabbages. There's a man in Southampton who grows herbs for us - wonderful oregano, marjoram, basil, fantastic varieties of thyme and sage, and Italian flat-leaf parsley. And there's a Sicilian farmer called Mario just off the M25 who grows piles of rocket and trevisano, winter leaves, broad beans..."

But did it have to be Italian? Would anyone be terribly distressed if you included Thai lemon-grass or Mexican beans or, I dunno, English rose petals in a dish if it made it taste more interesting. Didn't they get tired of Italian ingredients? "That's like saying, `Do you get tired of speaking English all day and would you like to speak a bit of French,' " said Ruth severely. Surely, I said, it's more like dropping the odd italicised French word into an English sentence in the interests of a lively style. She set her jaw. "We are committed totally to this Italian food experience, which is changing all the time and we will change with it. But I don't think we'll ever plant our own concept on it." "And anyway," put in Rose, diplomatically, "Italian cuisine has so many surprises. It's an inspiration, to go to a part of Italy and find that they use cinnamon in tomato paste. Or going to Capri and finding a salad of boiled lemons and artichoke. Even things that seem taboo to their cooking - like coriander, which you'd think would be strictly Oriental. It turns up in Italian food because it came up the African coast..."

How they love talking about food, even just naming the names - and with what rapt and greedy relish they taste everything on everybody's plate. My lunch with Ruth and Rose was punctuated by a kind of crockery square dance, in which I swapped plates with Ruth halfway through, so she could try my mozzarella di bufela with chargrilled aubergines and I could feast on Ruth's wood-roasted asparagus with gull's eggs, olives and salted anchovies; and just as I was resolving that no-one was gonna take this away from me, I swapped plates with Rose to try her tagliatelli with breadcrumbs and marcarpone and was glad I did.

It's an odd trajectory that has brought these very different women to this perfectionist haven. Ruth is from upstate New York ("Way upstate," she says, "You know Kingston? Poughkeepsie? Woodstock?") where her father was a radically-minded doctor and her mother a teacher. After school in Vermont, she discovered London in 1968, got involved in helping draft- resisting young Americans and went out with an Oxford Rhodes scholar (tantalisingly, a year after Bill Clinton was there). Like Rose, she worked as a graphic designer, at Penguin Books, and discovered the world of European cuisine by dining out every night in Paris when her architect husband was designing the Beaubourg Centre.

Rose is from Surrey. Her father was a balloon engineer who died before she was born, in a ghastly accident in the house beside the balloon shed. "I have literally only just discovered this story," said Rose. "I saw his grave for the first time three weeks ago. Nobody spoke about it, and my mother used to pretend he died in the war. Perhaps because of having a secretive mother, I've always been very enquiring about my origins, about food and gardening." She studied fine art at Guildford, taught art at Shoreditch Comprehensive, then raised four children and learned her cooking skills at the family hob. For a time, she made crepes in the intervals of rock concerts at the Rainbow and other venues, then left for America where a friend invited her to be head chef in a new restaurant.

The girls met in the mid-Seventies - Rose was an old friend of the first Mrs Rogers and shared a passion for Italian food, largely through the influence of Richard's Trieste-born mother, Dada. "She was the first Italian cook I met in London," says Rose, "I was 18 and we were all students, and she used to cook food one had never eaten in one's life." It all came together in 1986 when Rogers bought the Hammersmith warehouses to convert into his architecture practice and designated the site of the present restaurant as an eaterie. "I looked through application from caterers, and they were just dreadful," remembers Ruth. "I thought the only thing worse than not having an eating place was having a bad one. So I called up Rose and we sat down over coffee in Drummond's in the King's Road one morning in 1986 and said `Shall we do it?' and that was that."

Ten years later, they've done OK. Apart from the drift of international chef-dom across their deep blue carpets, they were described by no less than the New Yorker as serving "the best Italian food in Europe" - that's including Italy - and their cook book has sat like a prize marrow in the bestseller lists since it was published. Their refusal to cook beef or veal (because they can't get convincing guarantees out of meat suppliers) has more influence on London eaters than any amount of ministerial reassurance. And to be given lunch there, between the metal counter housing their chefs and the huge window with the view of their herb garden, is at least one guarantee of true love or serious intent in these uncertain times. Where food, power, charm and fastidiousness are the ingredients, the River Cafe is a dish that's perilously close to perfection itself.