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Brexit: David Cameron to quit after UK votes to leave EU
German chancellor Angela Merkel expressed "great regret" at the outcome, and EU chiefs said they expected the UK to begin negotiations to leave "as soon as possible, however painful that process may be".
But Boris Johnson, the ex-London mayor and public face of Vote Leave who is now a frontrunner to be next prime minister, said there was "no need for haste" about severing the UK's ties.
He said voters had "searched in their hearts" and the UK now had a "glorious opportunity" to pass its own laws, set its own taxes and control its own borders.
Another leading Leave campaigner, Labour's Gisela Stuart said the UK would be a "good neighbour" when it left the EU.
The pound fell to its lowest level against the dollar since 1985 as the markets reacted to the results.
Flanked by his wife Samantha, Mr Cameron announced shortly after 08:15 BST that he had informed the Queen of his decision to remain in place for the short term and to then hand over to a new prime minister by the time of the Conservative conference in October.
He would attempt to "steady the ship" over the coming weeks and months, but that it would be for the new prime minister to carry out negotiations with the EU and invoke Article 50 of the Lisbon Treaty, which would give the UK two years to negotiate its withdrawal, he said.
"The British people have voted to leave the European Union and their will must be respected," said Mr Cameron. "The will of the British people is an instruction that must be delivered."
Bank of England governor Mark Carney said UK banks' "substantial capital and huge liquidity" allowed them to continue to lend to businesses and households.
The Bank of England is ready to provide an extra £250bn of support, he added.
The referendum turnout was 71.8% - with more than 30 million people voting - the highest turnout at a UK-wide vote since 1992.
Mr Farage - who has campaigned for the past 20 years for Britain to leave the EU - told cheering supporters "this will be a victory for ordinary people, for decent people".
Labour leader Jeremy Corbyn, who called for the UK to remain in the EU but was accused of a lukewarm campaign, said poorer communities were "fed up" with cuts and felt "marginalised by successive governments".
"Clearly there are some very difficult days ahead," he said, adding that "there will be job consequences as a result of this decision".
He said the point he had made during the campaign was that "there were good things" about the EU but also "other things that had not been addressed properly".
Lib Dem leader Tim Farron said Labour's leader had been "utterly gutless" in the way he approached the campaign.
And two Labour MPs have submitted a motion of no confidence in Mr Corbyn's leadership which may be debated and voted on by Labour MPs next week.
Mr Johnson and Mr Gove paid tribute to Mr Cameron as they addressed Vote Leave supporters in London alongside Ms Stuart.
Mr Johnson said the UK was "no less united... nor indeed any less European" following the decision to leave the EU.
Meanwhile, at a press conference in Edinburgh, Ms Sturgeon said a second Scottish referendum was "on the table" and that the Scottish government would prepare legislation to enable one.
The European Parliament is to hold an emergency session on Tuesday to discuss the referendum result.
On Twitter, EU Parliament president Martin Schulz called for a "speedy and clear exit negotiation".
But Leave supporting Tory MP Liam Fox said voters had shown great "courage" by deciding to "change the course of history" for the UK and, he hoped, the rest of Europe.
A less than united Kingdom
Mark Easton, BBC home editor
The EU referendum has revealed an ancient, jagged fault line across the United Kingdom. It is a scar that has sliced through conventional politics and traditional social structures, and it is far from clear whether the kingdom can still call itself united.
The referendum was ostensibly about membership of the European Union. But voters took it to be asking a different question: what kind of country do you want Britain to be?
Yesterday seemed to offer a fork in the road: one path (Remain) promised it would lead to a modern world of opportunity based on interdependence; the other (Leave) was advertised as a route to an independent land that would respect tradition and heritage.
Which path people took depended on the prism through which they saw the world.
Britain is set to be the first country to leave the EU since its formation - but the Leave vote does not immediately mean Britain ceases to be a member of the 28-nation bloc.
That process could take a minimum of two years, with Leave campaigners suggesting during the referendum campaign that it should not be completed until 2020 - the date of the next scheduled general election.
Once Article 50 has been triggered a country can not rejoin without the consent of all member states.
Mr Cameron previously said he would trigger Article 50 as soon as possible after a Leave vote but Mr Johnson and Mr Gove, who led the campaign to get Britain out of the EU, have said he should not rush into it.
They also said they wanted to make immediate changes before the UK actually leaves the EU, such as curbing the power of EU judges and limiting the free movement of workers, potentially in breach of the UK's treaty obligations.
The government will also have to negotiate its future trading relationship with the EU and fix trade deals with non-EU countries.
In Whitehall and Westminster, there will now begin the massive task of unstitching the UK from more than 40 years of EU law, deciding which directives and regulations to keep, amend or ditch.
The Leave campaign argued during a bitter four-month referendum campaign that the only way Britain could "take back control" of its own affairs would be to leave the EU.
Leave dismissed warnings from economists and international bodies about the economic impact of Brexit as "scaremongering" by a self-serving elite.
The CBI said many businesses would be concerned about the referendum result.
It said "the urgent priority now is to reassure the markets", but warned against "rushed decisions".