More than two years in the making, this fascinating series shows many occasions when it nearly did all go wrong for Mandelson. Early on in his time as Minister responsible for the Dome, he is filmed canvassing a group of schoolchildren for their thoughts on how to fill the huge exhibition space. "You can tell me your brilliant ideas," he pleads, only half joking, "and how you're going to save my career."
The New Labour anthem may tell us that things can only get better, but in Mandelson's case they didn't. At the end of 1997, Stephen Bayley resigned as a creative advisor to the Dome, saying that "unless there is decisive creative management, it could be crap." The hapless Minister faced jeering headlines about "Mandelson's Folly".
He confides to the cameras that at the time "I felt my own career was slightly shaky. People were looking at me differently... and thinking `Is this man going to survive?"' Of course, he wasn't - but his downfall had nothing to do with the Dome. (BBC cameras were on hand to record the moment, on 21 December 1998, when Mandelson, scaling the Dome, received a doom- laden pager-message about his ill-advised home-loan: "The PM wants a word.")
For all the frissons these moments of potential disaster provide, the documentary also tells a story of potential triumph. When Mandelson first goes inside the Dome, it is already taking shape, and he is exultant: "I feel like getting down and kissing the earth. I'm feeling rather emotional. It's worth all the stress."
By June 1998, the exterior of the Dome is completed on schedule, and Tony Blair is impressed when he surveys the work. Mandelson can barely suppress his delight. "There are too many people in Britain who have forgotten what it is to be great as a country, too many people who have lost their ambition... I think it's pathetic."
Robert Thirkell, executive producer on The Dome, has been, to some extent, won over by Mandelson's vision. He reckons the Dome is so widely slagged off because "the public don't have the conceptual framework to understand it. We're always sceptical of grand ideas. The exhibitions of 1851 and 1951 were reviled in advance, too. The British character hasn't changed since 1851. It's easy to kick any creative endeavour. But I admire the Dome conceptually because it represents the notion of spending money on something more than the material. It's like building a cathedral; it's not just about making cars or hi-fis.
"When I was younger, I remember the feeling that Britain was a has-been country, the sick man of Europe. Now we're a younger, more dynamic, forward- looking country. We're not mired in the Imperial past, thinking how we used to be the biggest and the best. The Dome symbolises that change."
Adam Wishart, the series producer, says that part of the problem with the public perception of the Dome is that "there's nothing to show while you're making it. When James Cameron was making Titanic, everyone assumed it would be a flop because no one had seen the finished product."
"Building the Dome is like producing a TV programme," Thirkell chips in. "While you're working on it, you think `How will these disparate bits ever make a programme?' Then, suddenly, in the last week, everything comes together to create a unified whole."
Thirkell's cameras have captured some hairy moments - for instance, the kerfuffle when it emerges that the pieces of fabric for the roof do not initially fit the frame. He admits that some of the 20,000-odd people working on the pounds 758m project "may mind" about their portrayal in the documentary, but "what they wanted to show was the incredible amount of hard work that went into the Dome."
The filmmakers have certainly achieved that aim. You never know, they might even inspire some people to go and visit the blessed thing.
`The Dome' is on BBC2 at 9.30pm on Thur