''I agree to take upon myself the role of forming a government,'' Livni said, immediately after which she urged Benjamin Netanyahu, leader of Israel's right-wing Likud party, to join the coalition.
Livni now has 42 days to form a governing coalition in order to avert snap elections that polls indicate would bring Netanyahu's Likud party to power.
Traditionally, the task of forming a government goes to the party with the most seats in the Knesset, in this case Kadima, which has 29 MPs in the 120-member parliament.
Livni has already begun talks with parliamentary factions that could be included in a future coalition, while at the same time pressing members of her own party to close ranks.
But shifting allegiances in Israeli politics have meant that there is no guarantee that she will be able to come up with the numbers to form a coalition government and thus avoid an early election.
Polls indicate snap elections would bring Netanyahu's Likud party to power [AFP]
The Labour party, the main partner in the current coalition, has sent mixed signals, having met Netanyahu over the weekend and called for either early elections or a "national emergency government".
Livni met Labour's Ehud Barak, the defence minister, hours after Olmert formally submitted his resignation and offered to make the party a "full partner" in a new coalition, according to Haaretz newspaper, quoting unnamed Livni aides.
The ultra-Orthodox party, Shas - which has frequently played the kingmaker - has vowed to leave any government that negotiates the future of Jerusalem, a key issue in efforts to reach a peace deal with the Palestinians.
Livni also faces tough challenges in seeking to unify Kadima after new rifts emerged in the wake of her narrow victory over Shaul Mofaz, the transport minister and ex-army chief.
Formed less than three years ago, the centrist party has itself been dogged by allegations of corruption.
Olmert's decision to resign after just 32 months in office followed months of pressure from supporters and rivals alike in the face of a string of graft allegations against him.
He has denied any wrongdoing.
The continuing political turmoil has cast a shadow over Israeli peace talks with the Palestinians which were revived only in November, further denting hopes of a deal by the end of this year.
Both sides remain deeply divided on core issues, including final borders, Jewish settlements in the West Bank, the fate of 4.6 million Palestinian refugees and the future status of Jerusalem.
Shas's refusal to negotiate the fate of Jerusalem could prove particularly tough for Livni, who is currently the lead negotiator for Israel.
The Palestinians want mostly Arab east Jerusalem, which Israel seized in the 1967 Six Day War, as the capital of their future state.
But Israel considers the entire city to be its undivided capital, a claim not recognised by many countries within the international community.