At first, I think he's game-playing, but he assures me his children are all baptised Catholics. "My parent were socialists, so I was not allowed to go to religion classes in school. And anything you're not allowed to do as a child, you're interested in later on. And I always thought Catholicism was a more sound religion than Protestantism." I remark that, for a film-maker, the iconography is great. "There were a lot of things that I missed in Denmark's state Lutheran church. In Catholicism, you have the Virgin Mary; there's no women in Protestantism. And then you have all the saints. It's great you can go to different people with different problems. The saints are very colourful - and stubborn . . ."
He assured me that he prays - "a lot" - although he has little time for the institutional church. Suddenly he segues into a routine about his new, wooden house. "There's a lot of oak trees, and this year is a real acorn year, so for a month, we have been listening to dunk-dunk-dunk-dunk on the roof, right through the night . . . Life is a cruel idea, which is not a very religious or even Catholic thing to say."
Shucks, I hazard, life's a Valley of Tears. "Yes, but why did God invent this cruelty? Each of these acorns has a little embryo inside, and none of them will ever become a tree." I struggle to keep a straight face. "And if any of them become a tree, they will only do it by killing a lot of other trees . . .
"I have this vision of the old forests that used to cover Europe, really beautiful, with trees that have blown down, and all these plants grow on them, a romantic landscape, but in the forest, everything is fighting for survival. It's interesting that the image most people choose for relaxation is an image of a lot of beings trying to kill each other. "But what does the Catholic Church say about this? Why should anybody be tormented, even before they are born almost? They can't be sinners, these acorns."
Reluctantly, I remind him of the doctrine of original sin, for humans at least. "Ah, but how about the acorns?"
"I suppose I'm not a good Catholic," von Trier concedes. "First, I'm divorced, which is not good." Around the time of Breaking the Waves, he separated from his first wife, with whom he has two girls, aged 12 and six. Now he has a new family with non-identical twin boys, aged three.
I lever the conversation back, with difficulty, to his early feature films, Element of Crime (1984) and Epidemic (1987), dark, atmospheric, literary metafictions. Again, he is offhand. "That was mostly because of my co-writer, Niels Voersel. I stopped working with him because I was going into this sentimental stuff, and he has no interest in emotions."
I ask about what I consider von Trier's masterpiece, Europa (1991), a beautifully stylised, political film of ideas, and romantic tragi-thriller, shot in luscious black and white and set on a train traversing a benighted Germany, immediately after the second World War. Von Trier grew up in a non-practising Jewish family. His parents were senior civil servants, his mother a communist, his father a social democrat. "There was a lot of talk about the war, and we often went to concentration camps, and graveyards where you put little pebbles on top of the gravestones."
A decade ago, his mother, on her deathbed, confessed that his father was not, in fact, his real father. "It had a larger effect than I would have thought. In fact, I just met my brother and a sister on this other side two weeks ago. My brother is playing jazz bass, my sister's 60, and he's 63. They're very nice people. They didn't know about me."
"I met my real father once, and he just told me that any contact should go through his lawyer, which was not what I expected. He died half a year ago."
When he was in film school, he adopted the "von" in his surname. "A few years ago, to be German and to be noble was considered to be in extremely poor taste in Denmark. I used it as an artist's name like von Stroheim and von Sternberg, who were both fake vons. But now that I know none of my DNA comes from that source, it's more fake than I could ever think, both the von and the Trier."
Then he rolls off the sofa and takes me to the busy canteen for lunch, where he insists we both have a shot of schnapps, despite the early hour - "better early than not at all," he jests. "Schnapps will make you a better person."
Vinterberg wanders past and von Trier is rather ungracious when I compliment Vinterberg's Dogme film, Festen. In his neurotic way, von Trier is still displeased that Vinterberg pipped him to the first Palme D'Or - an award he has been set on winning for years.
Von Trier is a man of contradictions. He has a fear of flying, yet relishes hang-gliding. "Phobias have nothing to do with what is dangerous or not, it's something inside your head."
Strangely, for a man who makes such excessively emotional films, he has been taking the great emotion-leveller, Prozac, for four years. "Prozac's a very good drug. When I began taking it, I was in such a poor state that I couldn't leave my bedroom. Mostly I took it because I had just remarried, and I thought: `Come on! she deserves better than to have a husband who just flies under the carpet'. I've had that panic all my life."
But how does he manage to make movies? "Films have never been a part of my phobia. Films are the place where I am in control. Phobias have always been about things which were out of my control, diseases or bodily functions or whatever. But I've never been afraid of anything to do with film."
Dancer in the Dark is at selected cinemas