"The moment I saw the structure, I thought that, if it’s right, I’m going to be famous for all my life," Professor James Watson tells an audience at Clare College. The lecture hall is so full that even the stairs and floor are crowded with spectators. A similarly large audience gathered to hear Professor Watson talk at the Cambridge Union last night. He was, it turns out, right about that structure.

The structure in question is the DNA molecule: the famous double helix, a beautiful twisting compilation of two interlocking strands with bonds strung between them like the rungs of a ladder, deduced by James Watson and Francis Crick in Cambridge in 1953. Their Nobel prize-winning discovery has become the stuff of scientific legend, perhaps the greatest change in biology since Darwin. For not only did it show the genetic structure, it also explained how genes are replicated: in short, how the essence of life is stored and passed on.

As the DNA structure has become famous, so has the story behind its discovery. It is the tale of two youthful, brash and arrogant researchers who stole a problem that was not theirs to solve and left a tide of angry colleagues in their wake.

According to Watson, he was the enthusiast, driven by a teenage ambition to discover the nature of the gene. Crick was the brilliant theoretician, who needed Watson to convince him to focus on the DNA problem. "Francis [Crick] was the first person I could really speak to," Watson says. "Francis was really a super-genius.... I moved into the world of, you could say, ordinary geniuses and when I met Francis I saw a super-genius. That is, someone who really could focus on ideas."

They were united by a strong scientific curiosity, as well as by other incentives. "Francis and I are both running against God, and the reason we are running against God is that he doesn’t exist! We wanted you to show that you could understand the world without God. It was a strong motivation."

Strong motivation was certainly needed, as their work was beset by complications from the beginning. Neither one was officially allowed to be researching DNA. And, by tackling the problem, they were violating an unwritten gentleman’s agreement: Maurice Wilkins and Rosalind Franklin of King’s College London had begun work on DNA first, and so it was ‘their’ question.

The breakthrough came when Watson saw a technical photograph made by Rosalind Franklin that, although Franklin had not realised it, gave the crucial parameters of DNA. "When we saw it [the solution] that Saturday morning, that was it," an 82-year old Watson tells me when we meet. "It was going from no knowledge to knowledge."

When they were awarded the Nobel prize, a decade later, Maurice Wilkins was included. But Rosalind Franklin was by then dead and so ineligible for a prize not given posthumously. Watson has never managed to entirely silence critics who claim that she was hindered by a misogynistic scientific establishment. But Watson is unrepentant, claiming Franklin missed the implications of her own photograph. "Rosalind did not deserve the prize. Generally you don’t get a Nobel prize for failing."

As well as Franklin, many others who worked with Watson and Crick found the pair hard to deal with. "They made a movie, years later," Watson says, "and I didn’t like the movie, because the whole point of the movie was Francis and I were racing for a Nobel prize. Well, we weren’t, we were racing to solve what we thought was an extremely important scientific problem."

"We knew that if you solved an important problem, there was a chance you’d get a job later... And I was thinking, [if I solve it] I won’t have to go to North Dakota or something like that, because I never said anything to make friends. I said things because I thought they were right. If you follow that course of life, you don’t have many friends."

Not making friends is a problem that has followed Watson throughout his life. After leaving Cambridge, he became Professor of Biology at Harvard and then Director of the prestigious Cold Spring Harbor Laboratory. He has seemed unable to avoid controversy, attracting censure for his comments about genetic engineering - "People say it would be terrible if we made all girls pretty. I think it would be great" - and obesity – "Whenever you interview fat people, you feel bad, because you know you’re not going to hire them".

"About once every three years I get into real trouble for saying something I think is just obviously true," he tells me gloomily. The culmination came in 2007, when he told a newspaper he was "inherently gloomy about the prospect of Africa" because "all our social policies are based on the fact that their intelligence is the same as ours – whereas all the testing says not really". He was forced to resign from Cold Spring Harbor Laboratory as a result.

When asked about this, he is cautious and hesitating, clearly making an effort to choose his words carefully. "I never intended to –," he pauses. "I don’t remember making the statement... and didn’t wish ever to be a public figure because that’s not my field. I never wanted to make that statement and it got me into a lot of trouble".

But even when contrite, he is unable to prevent himself from speaking his mind, "We’re supposed to say that everything is equal... that all cultures all equal. I don’t think so!" he tells the Union audience.

"I think political correctness has gone too far," he tells me. "One of the big issues is: are you allowed to say there’s a difference between men and women? And I think we should be, because there are some things women do better than men and vice versa."

What about saying that men do science better than women, I ask? More pauses, more hesitation. "All I can tell you is that one of my postdocs is now the first woman president of MIT so I have not discouraged women. I never met a woman like Francis Crick, but I’ve never met a man like Francis Crick."

When we turn to talking about the big questions in science, Watson becomes contemplative. "The big question was: what is genetic information? We came up with the answer... and then some people said science is done, this was the grand question and everything else would be rather trivial, but it wasn’t true. The big question is how is information stored in the brain... and no one knows how to answer it."

Almost sixty years after his seminal discovery, his ambition seems unabated. "I want to cure cancer within the next five years." Characteristically outrageous, it is hard to tell if he is joking or not. But later he grows serious. "I won’t make it [a discovery]. My role now is to get people to do things that should be done. I got Francis to work on DNA, and it paid off."