The roof, which looks so delicate? It is, relatively speaking. Technically, it too is a kind of dome - but a dome stretched into a circle like a doughnut. It might weigh 800 tons (478 tons of steel and 315 tons of glass) but think: that weight relates to a latticework made of seven and a half miles of top-grade shipbuilding steel (6,000 beams and 1,800 connecting pieces) made weathertight with 3,312 panels of glass - each one covered exactly 50 per cent with those ceramic "frits", but each a slightly different-shaped triangle, and thus unique. The roof's computer-determined geometry takes up all the irregularities of the old building with almost scary precision - it had to be exact to a tolerance of just three millimetres - about a tenth of an inch. And because it flexes a little with heat, cold and wind, it sits on sliding bearings which also serve to spread its weight evenly across Smirke's facades.

Structural gymnastics take place in the corners to hold it all together, though you'd have to scramble up onto the roof to see all the tension cables and so forth: what drives the aesthetics of the scheme is its "look, no hands" feel from the Court: it just appears to sit there. The old iron frame of the Round Reading Room could not take any extra weight, so there the roof is supported on new columns hidden beneath a new stone skin. This is all clever stuff by Foster's highly-regarded engineers, Buro Happold. Who are also responsible, as it happens, for engineering the Greenwich Dome, which (forget the stuff inside it) is another world first. And this roof is designed to last rather longer.

So now, three years and £100m later, the builders have moved out, wiping their feet on the mat as they leave. What they have left behind, built so faithfully to Foster's designs that at first you feel you have stumbled into a giant computer graphic, is a rediscovered urban space. The Queen Elizabeth II Great Court, the largest of its kind in Europe, is an extraordinary public rendezvous. A courtyard, yes, but rendered doubly surreal. First by that beautiful roof, leaping from the square edges of the courtyard to the circular dome of the Reading Room, its filigree structure perching as lightly as a bubble over the space. And secondly by the presence of the Round Reading Room, which - being previously surrounded by bookstacks - was never seen, or meant to be seen, from the outside and thus was architecturally naked, its rough structure exposed. It has now been given a smooth limestone facade by Foster as part of a cunningly streamlined museum extension.

Through the glass roof you see the clouds chasing past outside, but as if through a light veil: the power of the sun is diffused by pale green ceramic dots or "frits" fused to the glass. The Great Court - which with its restaurants and cafes and big shop stays open later than the rest of the museum so you can get the cultural vibe in the evenings - doubles as a huge gallery in itself, for monumental sculpture. There you find the ancient Lion of Cnidos (from what is now southwest Turkey), dating from 300 BC. Also - heightening the slightly hallucinogenic feel to the place - you find a genuine long-eared Easter Island statue, name of Hoa Hakananai'a, as brought back by the Royal Navy's surveyors in 1868. Early in 2001, these will be joined by a no less monumental contemporary sculpture by Anish Kapoor in the form of a polished steel ellipse. First, they have to work out how to get this huge piece through the doors. No, really.

Commanding this newly-revealed space is the curious anomaly of the Round Reading Room. Smirke the Elder had introduced high-technology fireproof construction into the main museum: Smirke the Younger made the Reading Room ceiling out of papier mache, now lovingly restored and re-gilded. It is said that an intellectual graffito in the museum's lavatories once read: WIPE THAT SMIRKE OFF YOUR FACADE. Still, it seems the brothers got on. And by all accounts, the original courtyard was a shadowy, gloomy place anyway, where the lawns struggled to survive. Even the Victorians, no slouches at high-tech glassy buildings and fresh from the triumph of the Crystal Palace, toyed with the idea of flinging a roof across it. But when the Museum ran out of shelf space - just as the new British Library is already running out of shelf space - it had to go. In the Victorian mind, this was not a courtyard lost, it was a library gained. And given the influence the place has had on generations of writers, who can argue?