JERUSALEM—Foreign Minister Tzipi Livni won the Kadima Party’s primary election for its leader Wednesday, 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 Palestinians will proceed.

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

Livni needed 40 percent 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, the 50-year-old Livni will replace Prime Minister Ehud Olmert as head of Kadima. 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, Livni had 48 percent and to 40 percent for Mofaz, party officials told Israel Radio.

“You fought like lions ...,” Livni said in a phone call late Wednesday 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 Thursday after vote counting was completed.

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. Olmert will remain as a caretaker leader until a new coalition is approved by parliament.

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

“I am really happy that Livni won because she is committed to the peace process,” said dovish Israeli lawmaker Yossi Beilin. “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, 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, Livni favors diplomacy over confrontation, even though she said last week that she has “no problem pulling the trigger when necessary.”

A victory by Mofaz would have raised serious questions about Israel’s involvement in peace talks with both the Palestinians 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, Cabinet minister Meir Sheetrit and former Shin Bet security service director Avi Dichter lagged far behind in the polls.

Joyce Amiel, a Kadima supporter in Tel Aviv, said she was voting for 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 in Tel Aviv, the usually reserved 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.”