The government is under pressure to fill a vacuum left when Cameron announced he would resign by October after Britain ignored his advice and voted to leave the 28-member bloc in last week's referendum.
Triggering a leadership battle which could draw in some of his closest advisers, Cameron urged ministers to work together in the meantime. But he also formed a separate unit, staffed by public servants, to help advise Britain on its departure and its options for a future outside the EU.
"Although leaving the EU was not the path I recommended, I am the first to praise our incredible strengths as a country," Cameron told parliament.
"As we proceed with implementing this decision and facing the challenges that it will undoubtedly bring, I believe we should hold fast to a vision of Britain that wants to be respected abroad, tolerant at home, engaged in the world."
Asked about the possibility of a second EU referendum, Cameron said the result of Thursday's vote must be accepted.
Graham Brady, chair of the "1922 committee" of Conservative lawmakers which sets the party's ground rules in parliament, said the group had recommended that the leadership contest should begin next week and conclude no later than September 2.
That recommendation will almost certainly be passed.
"Both the Conservatives and the country more generally really want certainty. We would like a resolution and we think it would be a good thing to conclude this process as soon as we practicably can," Brady told Sky News.
He said there should be no new parliamentary election before Britain had negotiated the terms of its exit from the EU.
Several Conservative lawmakers have urged leadership candidates to try to broker a deal quickly to make sure that any campaign is as painless as possible, and to avoid deepening divisions exposed during the referendum campaign.
"A leadership contest now is not in the interests of our country," said Justine Greening, international development minister. "It will mean our party focuses inward at the very time our country most needs us to focus outward."
Jockeying For Position
All eyes are on former London mayor Boris Johnson, the most prominent of the "Leave" campaigners and now bookmakers' favourite to succeed Cameron.
But not all party members back him and many are pressing for "Anyone But Boris", seeing his decision to back the Leave campaign as a betrayal of his former ally Cameron, according to media reports.
Cameron's spokesman said the prime minister would not endorse any candidate to succeed him.
Instead, Cameron urged unity both in government and in the country and had announced he had set up an advisory unit to help manage Britain's departure from the European Union and to make sure his successor has all the possible information necessary to decide the country's future.
"Clearly this will be the most complex and most important task that the British civil service has undertaken in decades. So the new unit will sit at the heart of government and be led by and staffed by the best and brightest from across our civil service," Cameron said.
For now, the priority was working together on government business, which some critics say has been all but put on hold since campaigning for the referendum began in February, and reassuring the many migrants who fear their status may change.
Cameron told parliament he would not put up with intolerance, after reports that migrants, particularly those from Poland, had been told by some Britons to "go home" since the country voted to leave the EU.
"Let's remember these people have come here and made a wonderful contribution to our country. And we will not stand for hate crime or these kinds of attacks," he said. "They must be stamped out."