SACRAMENTO, CALIF. ? The California Legislature on Tuesday became the first legislative body in the country to approve a bill allowing same-sex marriages, but the measure faces an uncertain future with Gov. Arnold Schwarzenegger.

There were loud cheers by gay-rights activists in the Assembly gallery as lawmakers voted 41-35 to approve the bill and send it to the governor. The Assembly had twice defeated similar legislation.

A spokeswoman for the Republican governor said Schwarzenegger believes the issue should be decided by the courts, not by his signature on legislation. A state appellate court is considering appeals of a lower court ruling that overturned California laws banning recognition of gay marriages.

“He will uphold whatever the court decides,” Schwarzenegger spokeswoman Margita Thompson said.

Gay-marriage supporters nonetheless rejoiced in the victory. They compared the legislation to earlier civil rights campaigns to eradicate slavery and give women the right to vote.

The bill, sponsored by San Francisco Democrat Mark Leno, had failed in the Assembly by four votes in June, but Leno was confident he could get it through on a second try after the Senate approved a same-sex marriage bill last week.

“Do what we know is in our hearts,” Leno said. “Make sure all California families will have the same protection under the law.”

Assemblyman Tom Umberg, a Democrat who abstained in the vote in June, said he was concerned about what his three children would think of him if he didn’t join those “who sought to take a leadership role in terms of tolerance, equality and fairness.”

But opponents repeatedly cited the public’s vote five years ago to approve Proposition 22, an initiative put on the ballot by gay marriage opponents to keep California from recognizing same-sex marriages performed in other states or countries.

“History will record that you betrayed your constituents and their moral and ethical values,” said Republican Assemblyman Jay La Suer.

Randy Thomasson, president of the Campaign for Children and Families, a conservative group opposed to the bill, said Schwarzenegger should announce immediately that he would veto the legislation.

“Schwarzenegger can’t afford to sign the gay marriage license bill,” he said. “He’ll actually become a hero to the majority of Californians when he vetoes it.”

Opponents of same-sex marriage are trying to qualify initiatives for the 2006 ballot that would amend the state Constitution to ban same-sex marriages.

California already gives same-sex couples many of the rights and duties of marriage if they register with the state as domestic partners.