The British Museum's redeveloped great courtyard
He will take up the appointment next August when the current director, Robert Anderson, steps down.
Mr MacGregor joined the National Gallery in 1987 and oversaw the completion of the Sainsbury Wing and the refurbishment of the whole main display.
The trustees of the British Museum gained the formal approval of Prime Minister Tony Blair before announcing the appointment.
Mr MacGregor, 55, was educated at Oxford and in Paris and Edinburgh, and has written a number of books on architecture and art history.
Last year, he presented the major BBC series Seeing Salvation, which, over four programmes, examined images of Christ in western art.
"I am very honoured to be given the chance to lead The British Museum, one of the great museums of the world," said Mr MacGregor.
"After 15 happy years at the National Gallery, I shall greatly miss the colleagues and the pictures at Trafalgar Square."
The chairman of trustees at the British Museum, Graham Greene, said Neil MacGregor had been "outstandingly successful" at the National Gallery.
He was confident Mr McGregor would lead the British Museum with "equal distinction" in the run-up to its 250th anniversary, in 2003, and beyond, he added.
The National Gallery will now advertise for a new director, with the closing date for applications in January.
The appointment follows hard on the heels of the museum's announcement of a series of cuts to help tackle a projected £3m shortfall in its income.
These include halting plans for an £80m study centre and a freeze on recruitment.
The museum is also slicing £1m from its building maintenance budget.
It blames a drop in government funding - which it says has fallen by a third in real terms - and fewer visitors from overseas.
Unveiling the cuts in October, the museum criticised the government's handout to establishments which had only recently decided to adopt a free admission policy.
However, a spokesman for the Department for Culture, Media and Sport said the museum would have benefited by £700,000 due to VAT changes for free establishments.
"It is for the museum to manage the funds available to it," he added.