Brexit: David Cameron to quit after UK votes to leave EU
Speaking outside 10 Downing Street, he said he would attempt to "steady the ship" over the coming weeks and months but that "fresh leadership" was needed.
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".
The public face of the Vote Leave campaign Boris Johnson - a former London mayor and now frontrunner to be next prime minister - 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 leaves 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.
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".
Source: bbc.com