"The Right Honourable David Cameron accepted Her Majesty's offer and kissed hands upon his appointment as prime minister," it said.
Cameron said that he and Clegg wanted 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," he said in his first speech as prime minister.
"I love this country and think it's best days lie ahead. Now we need to confront our problems and take difficult decisions."
Liberal Democrat: 57
George Osborne, a close friend and ally of Cameron, is reported to become the new Chancellor of the Exchequer, or finance minister.
Some in the finance industry have expressed doubts about Osborne because he is untested and takes over an economy emerging from the worst recession since World War Two.
William Hague, a former Conservative leader and one of the main negotiators with the Lib Dems, is reported to become the foreign minister.
The British sterling currency rose against both the dollar and the euro as Cameron gave his acceptance speech.
Markets had been impatient to see an end to the uncertainty thrown up by last Thursday's inconclusive election.
Brown announced his resignation as prime minister outside his Downing Street office just an hour earlier on Tuesday.
Charlie Whelan, a former advisor to Brown, said it was "a sad day" and that there would be no rush to elect a new Labour leader.
Brown's resignation came after talks between the Labour party and the Liberal Democrats on forming a government had broken down, leaving the Conservatives and the Liberal Democrats to agree the final details of creating a coalition.
Al Jazeera's Alan Fisher, reporting from Downing Street, said that it was clear a deal had been made between the Conservatives and Liberal Democrats.
"Obviously a deal has been done and that is why Brown has resigned. Cameron has said that he wants his ministers to fill positions within hours of him taking office," Fisher said.
"I've informed the queen's private secretary that it is my intention to tender my resignation to the queen," Brown said in an emotional statement outside Downing Street alongside his wife Sarah.
"In the event that the queen accepts, I shall advise her to invite the leader of the opposition to seek to form a government.
"I wish the next prime minister well as he makes the important choices for the future."
Brown's two young sons emerged after his comments and then Brown and his wife were driven to Buckingham Palace where he formally tendered his resignation to the queen.
The Conservatives took the most seats in the May 6 election but had failed to secure an outright majority leading to a hung parliament for the past five days.