JERUSALEM — Foreign Minister Tzipi Livni won the Kadima Party’s primary election for its leader today, TV exit polls said, putting her in a good position to become Israel’s first female leader in 34 years and sending a message that peace talks with the Palestinian Arabs will proceed.

Cheers and applause broke out at party headquarters when Israel’s three networks announced their exit polls gave Ms. Livni between 47% and 49%, compared to 37% for her closest rival, a former defense minister and military chief, Shaul Mofaz.

Ms. Livni needed 40% of the vote to avoid a runoff next week, and her supporters hugged each other and shed tears of joy.

If official results bear out the exit polls, as is likely, 50-year-old Ms. Livni will replace Prime Minister Olmert as head of Kadima. Mr. Olmert, the target of a career-ending corruption probe, promised to step down as soon as a new Kadima leader was chosen.

The actual count was reflecting the exit polling. With about one-third of the votes tallied, Ms. Livni had 48% and to 40% for Mr. Mofaz, party officials told Israel Radio.

“You fought like lions …,”Ms. Livni said in a phone call late today to her headquarters to thank supporters.

“You did an amazing thing, and I just want to do all the things you fought for,” she said. “I know you did it as friends, but, like me, you did it because you want this to be a better place.”

She was expected to address party activists tomorrow after vote counting was completed.

Ms. Livni will have 42 days to form a new ruling coalition. If she succeeds, she will become Israel’s first female prime minister since Golda Meir. If she fails, the country will hold elections in early 2009, a year and a half ahead of schedule. Mr. Olmert will remain as a caretaker leader until a new coalition is approved by parliament.

Nationally, polls show Ms. Livni roughly tied with Benjamin Netanyahu of the Likud Party. A new nationwide vote would likely turn into a referendum on the current effort to forge a historic peace deal with the Palestinian Arabs.

“I am really happy that Livni won because she is committed to the peace process,” an Israeli lawmaker, Yossi Beilin, said. “I think the right thing for her to do now is to form a coalition that wants to promote peace rather than a broad government with the right.”

Foreign minister since 2006, Ms. Livni is Israel’s lead negotiator in the peace talks and is a rare female power figure in a nation dominated by macho military men and a religious establishment with strict views on the role of women.

A former lawyer, army captain, and one-time agent in the Mossad spy agency, Ms. Livni favors diplomacy over confrontation, even though she said last week that she has “no problem pulling the trigger when necessary.”

A victory by Mr. Mofaz would have raised serious questions about Israel’s involvement in peace talks with both the Palestinian Arabs and Syria. His approach is seen as far less conciliatory than hers. Had he won, the Iranian-born politician could have become Israel’s first prime minister of Middle Eastern, or Sephardic, descent.

Two other candidates, a Cabinet minister, Meir Sheetrit, and a former Shin Bet security service director, Avi Dichter, lagged far behind in the polls.

A Kadima supporter at Tel Aviv, Joyce Amiel, said she was voting for Ms. Livni “mainly because she is a woman, even though her positions are not clear. We think she would do the best job. We want her to win.”

Casting her vote at Tel Aviv, the usually reserved Ms. Livni bubbled with uncharacteristic enthusiasm. She said she was pleased with the turnout at her polling station and urged people to vote.

“You can determine today what the character of Kadima will be,” Livni said. “You can determine today if you really have had enough of old-time politics. Come and vote, bring your children, and show them how you are changing the country.”

The Palestinian Arab Information Minister, Riad Malki, was hopeful that peace talks could succeed under Israel’s new leadership.

“We welcome the results of the election, and we are going to deal with any new prime minister in Israel,” he told The Associated Press. “We hope this new prime minister will be ready to … reach a peace deal with the Palestinians that ends the occupation and allows the establishment of an independent Palestinian state living beside Israel.”

Kadima extended voting by a half-hour tonight, apparently to give voters returning from work more time to cast ballots.

Israeli media reported that about 55% of the 74,000 party members voted, with a crush as the deadline approached.

The primary was Kadima’s first since the party was founded by then-Prime Minister Sharon in 2005. Mr.Sharon suffered a debilitating stroke in early 2006, and Mr. Olmert subsequently led the party to victory in elections.

Mr. Sharon set up Kadima as a personal bastion after his hard-line colleagues in Likud blasted his unilateral pullout from the Gaza Strip in 2005. It was widely predicted Kadima would disintegrate after his exit, but the moderate Ms. Livni’s victory appeared to give it a chance of survival.

Mr. Olmert is under police investigation over his financial dealings. But he has been pursuing peace talks with the Palestinian Arabs and has pledged to continue as long as he is in office.

However, both he and his Palestinian Arab counterparts now say they are unlikely to reach the American-set target date of year’s end for a final peace deal. Also, any agreement they might reach would not be implemented until Mr. Abbas regains control of the Gaza Strip, overrun by Islamic Hamas militants in June 2007.

An Israeli political science professor, Gadi Wolfsfeld, predicted Ms. Livni could use a peace deal to win a national election.

“If she comes to a tentative agreement with the Palestinians, why not run on that platform, which would be very good for her?” he said.