JERUSALEM (CNN) -- Israeli Prime Minister Ehud Olmert handed in his resignation to President Shimon Peres on Sunday, Peres announced.

Israeli Prime Minister Ehud Olmert formally announced his intent to leave office at a Cabinet meeting Sunday.

"It's not an easy or simple decision," Peres said, moments after Olmert formally resigned. "I wish at this opportunity to thank the prime minister for his service to the country and the people during many years of public service."

Olmert will remain Israel's interim prime minister until a successor assumes power either by forming a coalition in the current Knesset or through general elections.

Peres is likely to ask Foreign Minister Tzipi Livni -- who won by a narrow margin to lead the ruling Kadima party last week -- to form a coalition. She will have a little more than a month to do so.

Peres said that starting Sunday evening, he will meet with the leaders of Israeli political parties represented in the parliament, or Knesset, before announcing who he will appoint to form a coalition.

He said he will announce his appointment before he leaves for the U.N. General Assembly in New York on Monday night.

Dogged by allegations of corruption, Olmert formally announced his intention to resign Sunday at a Cabinet meeting.

"This was not an easy or simple decision," he said before the meeting.

There are many possible scenarios that could play out in coming months after Peres called on Livni, who won by a narrow margin to lead the ruling Kadima Party last week, to form a coalition.

The Labor Party, the largest of Kadima's partners in government, could pull out of the coalition, which could force early elections or force the government to take on new coalition partners.

If elections are called, Ehud Barak, former prime minister and Labor leader, could vie for the top spot, but polls have shown he may not have enough support.

Some observers think former Prime Minister Benjamin Netanyahu, head of the opposition Likud Party, is also a potential candidate for the office.

Whoever succeeds Olmert as prime minister will be handed a set of daunting challenges, including determining the fate of Israel's talks with the Palestinians, its indirect talks with Syria and its tough talk on Iran's nuclear aspirations.

If Livni replaces Olmert, she will be the second female prime minister in Israel's history. Golda Meir served from 1969 to 1974. Watch Livni win Kadima vote »

Livni, a 50-year-old who entered the Knesset fewer than 10 years ago, owes her Kadima Party victory to her reputation for clean hands in a party that lost Olmert to allegations of graft.

She told reporters Thursday that she intends "to bring together Kadima factions and to go on this new path together."

Livni is the chief Israeli negotiator with the Palestinian Authority as the two sides work toward a peace deal. She also refuses to be tied to the Bush administration's vision of a peace deal by the end of this year.

Meanwhile, police have recommended that Olmert be indicted on corruption charges.

Israeli authorities say Olmert, while serving as Jerusalem mayor and a government minister, asked various public organizations to cover the same expenses and pocketed the extra money.

In May, an American businessman testified that he gave cash-filled envelopes to Olmert, who denies any wrongdoing.

Olmert was Jerusalem's mayor from 1993 to 2003 and served in several Cabinet posts from 2003 to 2006. He took over as prime minister after a 2006 stroke left then-Premier Ariel Sharon in a coma from which he has never recovered.

CNN's Shira Medding contributed to this report.

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