But, of course, Hadid's magnificently original design - proof of Cardiff's claim to be "Europe's most exciting waterfront development" - now looks unlikely actually to be built. Heritage Secretary Virginia Bottomley's Christmas present to the city was the announcement that the application for an initial Millennium Fund grant of pounds 2.75m to eliminate uncertainties in the project had been refused. The Opera House was looking to the Fund for the bulk of the estimated total cost of pounds 86m. With that initial grant withheld, the prospects of the Opera House going ahead are slim.
So Hadid stands at the moment of her greatest fame, but the creative achievement to which she owes that fame now seems likely to wither on the branch; the satisfaction of seeing her vision realised is to be snatched away. Instead she must endure the crowing of the fogeyish critics for whom her work symbolises what is most arrogant about modern architecture in Britain. She has the consolation of having been short-listed to design a new block for the V&A, and has earnt the right to minor-celeb treatment - photographers fussing round her flat with reflectors and tripods, magazine features on the decor. But it's a poor substitute for getting something built.
THE FLAT in question is on the first floor of a grand, stucco-fronted Victorian house, overlooking a leafy square, near the Cromwell Road in west London. The large front room is dominated by two outsize sofas of her own design. There are angular Hadid coffee tables and a spectacular collection of glassware, but otherwise, as in many architects' homes, there is an exaggerated bareness about the place - no books, no shelves to put them on. You know there has to be a backstage cubbyhole somewhere that you are not going to see, crammed with possessions.
She is still riding the backwash from the Millennium Fund's decision. "People are just upset about it," she says, frothing milk for cappuccino in her small, neat kitchen. "People you wouldn't expect - young people especially - say to me, 'If this building can't happen, what can?' I've had an incredible reaction from abroad about it ..."
This is not the first time ill luck has befallen one of her projects: in fact her career has been punctuated by similar misfortunes. It's an architectural version of the torments of Tantalus: a competition is held; against all the odds Hadid wins; the photographers, the 15 minutes of fame ensue; then before one spade of earth can be turned, fate intervenes and kills the thing off.
The first time this happened to her on an important project was in 1983, in the case of the Peak, an exclusive members' club and private residence in Hong Kong. There it is on the wall above us as we sip coffee, the large semi-abstract painting which is her "storyboard" version of the building: jumbo jet coming into land at Kai-Tak airport, the fat concrete beams that were to have been the building's horizontal superstructure flying the opposite way, heading for the building site. As at Cardiff, Hadid came from outside to win the competition against a formidable international field, but when all the hoopla was over, the building upped and died. She still doesn't understand what went wrong. "All I know is that I won at exactly the time when they decided Hong Kong would go back to China. The competition was announced before Thatcher went to China, and I won it after she went." That's the sort of bad timing a budding career doesn't need.
Yet despite never being built, the Peak was the key moment in her progress: the point at which Hadid became impossible to ignore. The daughter of an LSE-trained Iraqi economist, she was brought up in privileged circumstances in Baghdad, rounded off her education in England, Switzerland and Beirut, then came to London 25 years ago to study at the Architectural Association, the world-famous architecture school in Bedford Square. She hit this establishment - still a vibrant and trailblazing school today - when it was at a particularly heady moment. In the Seventies, the range of approaches and ideas at the AA was startling: Hadid's teachers included Leon Krier, later to become Prince Charles's architectural guru and the moving spirit behind Poundbury, his artificial village in Dorset; and Rem Koolhaas, the Dutch ex-journalist and screenwriter, whose book Delirious New York, a hymn of joy to skyscrapers and congestion, has recently been republished. "There were so many diverse people," Hadid says now, "that, although it created a degree of confusion, it really made you focus on what you wanted to pursue."
Her path took her initially into the past, to root around among the paintings of the Suprematists, the artists who sprang up in Russia in the hectic years before the October Revolution, and painted febrile visions of the architecture of the future. Although the Suprematist spirit still infuses the paintings with which she illustrates her designs, Hadid made no attempt to imitate what she found there. These fantastic visions were merely her departure point. The key idea, which was to make possible all the works that followed, including Cardiff, was fragmentation: shattering conventional modern forms into eccentric bits, then piecing them back together in new ways.
Why should this curious procedure have occurred to her? The answer is that it was in the air: the French philosopher Jacques Derrida had caused a sensation in the late Sixties with what he called the "deconstruction" of literary texts - tearing them apart to examine what they were made of, shining a light on their unconscious premises, trying to discover what the author was really up to, beyond his conscious intentions.
Hadid was one of a number of architects driven to attempt something similar in their own field. Classical modernism had grown stale and boring and was increasingly unpopular: its repertory of pure geometrical forms, supposedly pointing to an ideal world of truth and purity, could no longer be taken on trust. Instead it was ripe for subversion, for shattering. If you took these squares and rectangles of glass, concrete and steel, and smashed them into hundreds of fragments, what might you learn? What new thing might you be able to make?
THE ONLY place in the world where you can see Hadid's work as she herself intends it to be is outside the German town of Weil am Rhein, near the Swiss and French borders, where she built a fire station for the protection of the chair factories of the Swiss company Vitra.
The chief executive of Vitra, Dr Rolf Fehlbaum has adorned his factory with the work of several of today's most interesting architects: Nick Grimshaw (of Waterloo Terminal fame) did the basic plant buildings, Frank Gehry the museum, the Japanese Tadao Ando the conference centre. Hadid's fire station is tucked away at the rear of the site. It performed successfully as a fire station for two years after construction but will soon be converted into an annexe of the museum for architectural exhibitions. (Vitra now relies on the municipality's firemen.)
The chief characteristic of the fire station is dynamism: like one of Hadid's drawings, it seems to be in the process of tearing itself apart and flying off in all directions. The building is dragged out, long and thin, along the longitudinal axis; its forms - not only those of the structure itself but also of the sinks and free-standing lockers within it - seem to be in the grip of forces that are wrenching them out of shape. The building, more like a habitable sculpture, conveys a sort of visceral excitement that is not uncommon in large dramatic structures, such as Grimshaw's Waterloo Terminal, but unthinkable in most buildings this small.
To describe Hadid's only fully realised work thus is to risk giving the impression that the Cardiff experience would be similar. But the fire station's aspiration is modest; Cardiff is an infinitely more ambitious scheme.
The conundrum that has troubled modern architecture ever since it began to tire of such "perfect forms" as Mies van der Rohe's curtain-walled boxes is: where do we go from here? Having achieved a form this "pure" - what next? With her compulsive smashing and sticking together, Hadid has in her Cardiff design found an answer.
A building like the Opera House has to do many different things: stage operas; accommodate rehearsals, offices, workshops, car parking, restaurants and shops; present visitors with an arresting, memorable image; offer views of the bay; strike up friendly conversations with neighbouring buildings (which don't yet exist); attract a large number of visitors and keep them amused. As the judges of the competition discovered, for a single monolithic form - Sir Norman Foster's half-dome, for example - to perform all of these functions is a tall order. It might manage half of them; the rest fall by the wayside.
What Hadid sacrifices in her approach is the monolithic quality that for a long time was inseparable in people's minds from architectural modernity - think of the Pompidou Centre, or the Royal Festival Hall. Cardiff, like the fire station, will never have "integrity" of this literal sort. It is fragmentary - but for that very reason can do a dozen different things. Need a strong image? The mighty auditorium, set in the middle, will be visible for miles. Need to give office workers light and air, while insulating the music spaces from outside noise? Wrap the linear office accommodation, "the necklace" as Hadid calls it, around the auditorium; lodge rehearsal spaces here and there in its inner circumference. Want to free the ground so people can walk around on it? Partially hoist the necklace blocks up into the air ... and so on. The final result, were it to happen, would be the bustle of disparate activities, the serendipity of intersecting views, activities and pleasures that means the essence of a lively city, when it's working well.
Debate about the Opera House has flared up repeatedly over the years of the project's history. Hadid's design won a convincing victory 14 months ago; but when local opinion, canvassed with ludicrous superficiality by television, voted by a large majority against it, some of the trustees took fright and tried to find a "safer" winner from among the rejected schemes. But Hadid stood her ground and improved her scheme further, and was eventually back in the driving seat.
With the Millennium Commission's thumbs-down in December - for reasons that have not been clearly explained but are thought to be a mixture of financial worries, fear of tabloid-inspired fury against Lottery money going to "toff" culture, and nerves about the design - the knives were out for Hadid again. Yet all is not lost: the National Museum of Wales wants to get involved, changing the project's nature from an opera house to a more broadly based cultural centre. A revised proposal incorporating this idea is due to be put to the Millennium Commission tomorrow.
It is not too late for them to change their minds. If they do, they will endorse the only new urban building, monumental in both scale and ambition, which will be ready in time to greet the new millennium. Not only for the sake of Wales but also for the sake of architecture, the thumb should go up.
! Zaha Hadid talks about her work at the ICA, SW1 (0171 930 3647) at 7.30pm on 22 Feb.Reuse content