Earlier in the day Cameron promised that the alliance would be a "full and proper" coalition between the two parties.
The Liberal Democrats, who came third in last week's inconclusive election, have been given five cabinet seats, with Clegg taking the role of deputy prime minister.
The new government is hoping to reduce Britain's record $236bn deficit, as the European Union highlighted the need for "common responses" to economic crises.
George Osborne, the new finance minister, said the government planned "long-term structural reforms of the banking system, of education and of welfare so that we have an economy that works for everyone".
Liberal Democrat: 57
The coalition was formed after no party managed to gain a clear parliamentary majority in the May 6 election.
But Cameron, whose party won the most seats in the poll, announced his party would be joining forces with the Liberal Democrats after Gordon Brown resigned from his post as prime minister on Tuesday evening.
Brown stood down after Labour failed to clinch a deal with the Liberal Democrats.
Cameron said the new alliance would not just be a new government but a "new politics".
"Working together, I know we can take the country through those difficult
decisions to better times ahead.
"But today we are not just announcing a new government, and new ministers. We are announcing a new politics."
Alan Fisher, Al Jazeera's correspondent in London, said the coalition would provide a "different way of looking at politics".
"No more is there going to be this Punch and Judy style of politics - there will be compromise.
"Maybe this is a way of looking differently at politics. But the bookmakerss and Labour are giving them no more than two years at the most."
A number of cabinet positions were announced on Wednesday, including William Hague, a former Conservative leader, to serve as foreign minister and Liam Fox as defence minister.
Theresa May, previously shadow work and pensions secretary, has been appointed as interior minister.
Vince Cable, the Liberal Democrats economic spokesman, will become the government's banking minister and Chris Huhne, another Liberal Democrat MP, will take on the role of environment secretary.
Markets had been impatient to see an end to the uncertainty thrown up by last week's election and Britain's sterling currency rose against the dollar and the euro as Cameron spoke.
But some in the finance industry have expressed doubts about Osborne, an untested cabinet minister, becoming chancellor at a time when the economy is emerging from the worst recession since the Second World War.
Barack Obama, the US president, was one of the first world leaders to congratulate Cameron on his appointment, affirming the two countries' "special relationship".
The leaders of Germany and France also congratulated the new prime minister, while the European Commission underlined the need for common responses to economic challenges in a congratulatory note to Cameron.
|Clegg, right, is considered a bigger supporter of the EU than Cameron, left [Reuters]
Many European leaders are wary of the prime minister's centre-right Conservatives coming to power because they are more hostile to the 27-country European Union than the outgoing Labour party.
Jose Manuel Barroso, president of the EU executive, said Cameron's government faced difficult choices in difficult times but he was confident it would chart the right course to steer Britain out of crisis and towards sustainable growth.
Cameron is widely expected to want to show more eurosceptic members of his party that he will defend Britain's interests strongly in the EU.
But the new government could face internal problems over Europe, as Clegg, once a member of the European parliament, has long been a supporter of the EU.