The centre-left leader, however, still faced last-minute negotiations with his allies over cabinet posts.
Giorgio Napolitano, the Italian president, gave Prodi the mandate after consulting former presidents and party chiefs, including Silvio Berlusconi, the outgoing centre-right prime minister, who insisted he wanted more checks on the results of the election.
Prodi accepted the mandate and continued talks with his coalition allies, who range from Roman Catholic moderates to communists, over the make-up of the government.
"I think it's going to be quite a long night," he said shortly after leaving the Quirinale presidential palace.
Prodi said he expected to have the final cabinet list by Wednesday morning. The government is expected to be sworn in on Wednesday.
Ministries including interior, defence and justice were still undecided as of Tuesday evening.
Days of bickering over cabinet jobs have underscored the problems Prodi is likely to face with a wafer-thin majority in parliament and Berlusconi's pledge to lead a "an opposition without pity".
His government needs to get down to work quickly to try to revive a stagnant economy, with rating agencies threatening to downgrade Italy's debt if badly needed but unpopular reforms are not carried out soon.
Berlusconi wants parliament to
launch a probe into some ballots
Napolitano said no one in Berlusconi's coalition had contested that Prodi should be named prime minister, but the media tycoon said he wanted parliament to launch an investigation into ballots from Italians abroad and those classed as void.
Berlusconi later said members of his centre-right bloc should chair the committees in charge of the probe.
"As you know, there were many irregularities, many anomalies that we want cleared up," he said in his final news conference as prime minister after a record five years in office.
Prodi's mandate was delayed, first by a recount of disputed ballots and then by the election of a new president.
Once Prodi's team has been announced, he must still win a confidence vote in the Senate, probably by Friday, and another in the lower house early next week, before taking office.
Prodi appeared to have decided on two deputy prime ministers - Massimo D'Alema, chairman of the Democrats of the Left party who is also tipped to be foreign minister, and Francesco Rutelli, leader of the centrist Margherita (Daisy) party.
Tommaso Padoa-Schioppa, a former European Central Bank board member, looked certain to be named economy minister.