The PM had urged the country to vote Remain but was defeated by 52% to 48% despite London, Scotland and Northern Ireland backing staying in.
UKIP leader Nigel Farage hailed it as the UK's "independence day", while Boris Johnson said the result would not mean "pulling up the drawbridge".
Scottish First Minister Nicola Sturgeon said she was "absolutely determined" to keep Scotland in the EU so a second Scottish independence referendum was now "highly likely".
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 front-runner 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.
The slide halted when the Bank of England said it stood ready with £250bn of extra funding to ensure market stability. But the pound remains 8% down against the dollar, and 6% lower against the euro.
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.
Chancellor George Osborne said he had briefed G7 finance ministers about the consequences of the vote.
On Twitter, he said it had been a "hard-fought campaign" and while it was not the outcome he wanted, "I will do all I can to make it work".
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".
There have been calls for him to consider his position after some of the strongest votes to leave the EU came from traditional Labour heartlands.
Senior backbencher Dame Margaret Hodge has tabled a motion of no confidence, which has the support of eight other Labour MPs, urging Mr Corbyn to resign. The motion could be debated and voted on by Labour MPs next week.
Lib Dem leader Tim Farron said Labour's leader had been "utterly gutless" in the way he approached the campaign.
But major unions, which fund the party, have urged Labour MPs not to create a leadership crisis.
Shadow chancellor John McDonnell said his party needed to "gear up" for a possible snap election and that he was "disappointed" at Dame Margaret's intervention.
http://www.bbc.com/news/uk-politics-36615028
KEYWORDS:
1. drawbridged 吊橋
2.referendum 頃示書
3.halted 終止
4.marginalised 排斥
5.backbencher 後座議員
6.utterly 徹底地
