Here, in one Joycean passage, is Branagh on having a private life that, like a stately home, is unavoidably open to public view: "Given that the lines of demarcation around celebrity now are very blurred, I think you have to make your own decision about where it begins and ends, so while some of it is out of your control - people can write or say what they want - you don't have to be any more complicit with it than you choose to be, so you can invite more or less of it, or you can just decide to draw your own line in the sand and it doesn't have to be a big dramatic gesture - there are some things you talk about and some things you don't, and if people still want to write about them, that's fine, and you understand the appetite and all the rest of it, but I kind of - my view is that if I'm not interested in person X's private life, you know, I don't necessarily feel that I have to share mine ... except that given the job I do I have to expect that some people will possibly write about it, accurately or not accurately but, uh ... I don't uh ... you can't become an actor and worry about becoming a victim in that way."
Branagh's subject here - the cult of celebrity - is also the subject of the new Woody Allen film, Celebrity, in which he takes the role that would normally be played by the bespectacled one himself. Branagh is Lee Simon, a journalist who dabbles in celebrity interviews and features for travel magazines while struggling to finish a novel. Many colleagues will allow themselves a little shiver of recognition here. (In my own case, the first-name coincidence caused an extra frisson.) On the emotional level, though, Lee Simon is like no one so much as Woody Allen - or rather, the Woody Allen persona we have come to know and love/hate in successive films. Recently divorced from the insecure Robin (Judy Davis) because he "needs more space", Lee is a man who hides his selfishness, especially in relation to women, under a veneer of endless verbal self-justifications and appeals to romantic - or sexual - impulse. A witty mess of complexes, in other words, but still a bastard.
But the character Branagh is playing is overshadowed by his take on the part. He doesn't just put on a New York accent, or attempt to method-act his way into the mind of the successful but frustrated journalist. He becomes Woody Allen - down to the smallest inflection, the slightest gesture. For a while, as with any good impersonation, this is mesmerising, but in the end it detracts from our belief in the character. Oddly, it was only when I watched the film dubbed into Italian that I was able to see Lee Simon as anything more than a take-off.
So why he did he choose to impersonate Allen? "His comic voice is so specific - unique - that it's pretty much impossible to play against it." But others have managed - John Cusack, for example, in Bullets over Broadway, one of the few Woody Allen comedies in which the director did not play the lead. Pressed, Branagh suggests it was Allen himself who insisted on such a close reading: "He directs, sometimes, very specifically. He'll give you a line reading and he'll do it and you'll do him, copy him."
Alongside the serious Shakespearean actor, there has always been a streak of the schoolboy mimic in Branagh. He loves showing the Americans that he can do their accents as well as they can. In the John Grisham adaptation The Gingerbread Man, which was released last year to mixed reviews, he played a lawyer from deepest Georgia. He has told the story of walking into a bar in Savannah and asking for a glass of wine in his on-set southern drawl. "Why're you talking like that?" the barman rebuked him. "You're that Shakespeare guy, right?"
As his readiness to share this anecdote shows, there is a strong element of self-deprecation in Branagh's manner. Which perhaps explains why he has consistently foiled those critics who expected the Great White Hope of British cinema to manage his career with De Niro-like discernment. In the tidal wave of adulation that followed the release of Henry V in 1989, the 28-year-old Branagh became, briefly, a sort of up-market (and very British) Leonardo DiCaprio. His picture was not on quite so many bedroom walls, but his private life - including that marriage - was just as wide open to public scrutiny. Now, 10 years on, Branagh finds himself playing alongside the Titanic boy wonder in Celebrity's highlight sequence - a manic afternoon in the life of a brat-pack star, complete with cocaine, private jets, roulette wheels and four-in-a-bed sex. Of working with the 25-year-old DiCaprio - a few weeks before Titanic was released - Branagh says that "all the people in that group decided they were being mad and bad, so it was all very larky. I felt about 400 years old anyway, with these slips of things".
These days, Branagh's own professional life is more of a steady downpour than a tornado. After his unadventurous outing in last year's period thriller The Proposition, he came back briefly to London to make The Theory of Flight, in which he plays the part of an artist and inventor obsessed with flying machines. The role of Jane, the young girl in a wheelchair he is assigned to care for when one of his experiments ends him up in court, is taken by his current partner, Helena Bonham Carter. (It's their second film together; they met on the set of Mary Shelley's Frankenstein.) A BBC production, the film has so far been seen almost everywhere except on home ground. Ostensibly this is to do with the summer log-jam of Branagh features: next up, in August, is the comedy western The Wild, Wild West, in which he stars alongside Kevin Kline and Will Smith. But a lukewarm reception from both critics and audiences in the States, where The Theory of Flight was released in January, has done nothing to move the film further up the UK schedule.
It's easy to knock Branagh for his career choices. One problem lies not so much in his competent but hardly earth-moving Hollywood performances as in our (probably misplaced) expectations of him. After all, if our yardstick were, say, Keanu Reeves rather than Laurence Olivier, Branagh would be up there with the greats. The other problem is that, unlike Reeves, he cannot rely on staring moodily into the middle distance. So he is forced to act - and, given his apparent inability to turn in a flat, cinematic, uninflected performance, this generally means overacting. He is at his best, in fact, when the role requires overacting - as when he played Iago in Oliver Parker's 1995 film of Othello. And his self-directed Shakespeares have all seen him rise to the occasion, even the overly self-conscious Hamlet.
It wasn't just audiences that were put to the test by this four-and-a- half-hour marathon. Branagh admits that the job of financing and making Hamlet was "immensely effortful ... I ran out of energy, anyway - I was well f---ed by the end of that. And I felt I'd rather not have that total responsibility for a while".
Hence the undemanding Hollywood roles - coming to the surface and catching up on the tan between bouts of full Shakespeare immersion. But there is a risk in taking such a breezy approach to one's filmography - the risk being that the general public, encouraged by the sort of "Who's in, who's out" journalism that Lee Simon embodies in Celebrity, will see Branagh as someone who has simply lost control of his career. After all, this is the man who once said "my definition of success is control".
Being confronted with former quotes is almost as bad as being confronted with former girlfriends, and Branagh now considers this to be "a stupid remark". So what would be Branagh's current definition of success? "It's some sense of creative freedom when it comes to directing," he says. "The freedom to pursue those projects you care very passionately about, and then the freedom to do them in a way that you believe in, and not be confined by, these days, the incredible pressures put on film-making, from casting to budget to the marketing to the preview process and all the rest of it ... you somehow find a way to have your voice in that. That's success for me, really".
The project he cares very passionately about at the moment is Love's Labour's Lost, his first stab at Shakespeare in the three years since Hamlet. Couched in the form of a 1930s musical, with songs courtesy of Cole Porter and Irving Berlin, it will be released in the autumn.
Every actor, they say, ends up in a Woody Allen film sooner or later. Plenty of actors have also queued up to do Shakespeare with Ken. Now Branagh is in a Woody Allen film. Surely the logical next step would be ... but this, it seems, has already been tried. "I once had the idea of asking Woody Allen to appear in one of my films. After a while I got a very politely worded letter back from him asking me what I'd been smoking when I made the request."
'Celebrity' is released on Friday