Mr Cameron said the country faced deep and pressing problems but that he and Nick Clegg, the Liberal Democrat leader, would put aside party differences to deliver strong government.
He said it would be “hard and difficult work” to govern as a coalition.
“I aim to form a proper and full coalition between the Conservatives and the Liberal Democrats,” he told reporters in a televised statement.
“I believe that it the right way to provide this country with the strong, the stable, the good and decent government we need so badly.”
Mr Cameron paid tribute to Mr Brown’s for his “long record of dedicated public service” and said that after more than a decade of Labour rule, Britain was “more open at home and more compassionate abroad”.
Looking ahead to the coalition he will form with the Lib Dems, he said: “We have some deep and pressing problems – a huge deficit, deep social problems, a political system in need of reform.
“For those reasons, I aim to form a proper and full coalition between the Conservatives and the Liberal Democrats.
“I believe that is the right way to provide this country with the strong, the stable, the good and decent government that I think we need so badly.”
He added: “Nick Clegg and I are both political leaders who want to put aside party differences and work hard for the common good and for the national interest.”
“I believe that is the best way to get the strong Government that we need, decisive Government that we need today.
“I came into politics because I love this country, I think its best days still lie ahead and I believe deeply in public service, and I think the service our country needs right now is to face up to our really big challenges, to confront our problems, to take difficult decisions, to lead people through those difficult decisions so that together we can reach better times ahead.”
Earlier, Mr Brown made his announcement in an emotional speech outside Downing Street, where he said “thankyou and goodbye”.
The former PM’s dramatic resignation came a day after he announced he would step down as Labour leader in the wake of the party’s defeat in last Thursday’s General Election.
After five days of uncertainty, Mr Brown finally accepted that he was unable to form a coalition with the Liberal Democrats.
After five days of uncertainty, Mr Brown finally accepted that he was unable to form a coalition with the Liberal Democrats.
The pound rose toward the strongest level in almost a year and gilts gained within minutes of Mr Brown’s announcement.
