Construction on Crossrail is due to start in 2010
Crossrail graphics
Construction for the link - from Maidenhead, Berkshire, through to Essex - is expected to start in 2010.
It will provide 24 trains an hour into the heart of London from the east and west, improving rail links to the West End, the City and Docklands.
Mr Brown said it was a great day "for London, Crossrail and the economy".
He said the project was of "enormous importance, not just for London but for the whole country" and would generate 30,000 jobs.
The government is providing a third of the money with the rest made up from borrowing against future fares and a levy on London business rates.
See details of Crossrail route through central London
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Mr Brown said: "I believe we have come up with a workable, robust and cost-effective business plan.
"Crossrail will add £20bn to the UK gross domestic product. Today's go-ahead proves that London works best when London works together."
Transport Secretary Ruth Kelly said the project had "eluded all previous governments" due to the funding gap and the scheme would leave a "lasting transport legacy".
Crossrail services are due to be running in about 10 years.
The idea was first proposed in the 1980s, but supporters have had difficulty in securing the funding.
'Giant jigsaw'
Describing the move as an "internationally recognisable vote of confidence" in London's economy, Mayor Ken Livingstone said: "Crossrail is not just a transport scheme, it is the key to the next 20 years of economic development of London."
Baroness Valentine, chief executive of business group London First, said it was like a "piecing together a giant jigsaw" and would be crucial "when Chinese and Indian businesses consider where to site their European HQs".
Shadow transport secretary Theresa Villiers welcomed the decision, but asked: "Why has it taken Mr Brown 10 years to commit to the funding?
"Every time an election seems imminent, the Government wheels out the promise of Crossrail. I hope that Brown is really on board and not using it as another political stunt."
Her sentiments were echoed by Liberal Democrat transport spokeswoman Susan Kramer.
She said: "It is great news...however, this process has dragged on for far too long. It is a shame that it has taken the possibility of a general election to make things happen."
The news was also welcomed by the Rail Maritime and Transport union (RMT), who urged the government to go further by approving a new north-south high-speed rail link as well.
As well as adding capacity to London's overcrowded tube network, Crossrail will improve links to Heathrow and other airports.
Earlier this week, the City of London Corporation agreed to "a financial contribution" to help meet a funding gap of £400m that was holding back the scheme.
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The Crossrail Line
The £16 billion Crossrail Scheme, that will run through London, Berkshire and Essex was given the go ahead; construction will begin by 2010 and it is expected to be finished by 2017. The line, that will provide 24-hour trains to the centre of London from the east and the west, will provide a better rail link to the West End.

UK Prime Minister Gordon Brown said "It was a great day for London, Crossrail and the economy" with the project expected to add £20 billion a year to the UK economy.

The government is providing a third of the money with the rest made up from borrowing against future fares and a levy on London business rates.

The news was also welcomed from the railway union RMT, who had been protesting to the government to go further for a new north to south high speed rail link.

Crossrail is also hoped to reduce the overcrowding to airports such as Heathrow and many others.