Read local coverage at CNN affiliate WXAA in Albany, New York.

(CNN) -- New York's state Assembly Wednesday night approved a same-sex marriage bill, sending it to the Senate, where it faces a closer vote.

The Marriage Equality Act passed the Assembly 80-63.

It would grant same-sex couples equal rights to marry "as well as hundreds of rights, benefits and protections that are currently limited to married couples of the opposite sex," according to Gov. Andrew Cuomo's office.

"This is an immense step toward achieving true equality for all here in New York," said Democratic Assemblyman Daniel O'Donnell, who argued that he and his partner should be allowed to marry.

Wednesday night's debate included discussion of religious, traditions, customs and families. Small gay rights flags decorated many desks in the chamber.

Same-sex marriage is a fundamental human right, said Democratic Assemblyman Richard Gottfried during the debate.

Democratic Assemblyman Micah Kellner also supported the bill, saying, "This is about people's happiness."

Republicans defended their opposition, saying it did not mean they were intolerant.

"It doesn't make me a bigot," said Assemblyman Steve McLaughlin. "It makes me a traditional American."

"Marriage to me has a separate importance and it has to do with a man and a woman," said Republican Nancy Calhoun.

Senate Republicans also met Wednesday on the issue, and discussed the law's wording.

There was no decision on when it may come to the floor. A vote there likely won't happen until Friday, at the earliest.

The state Senate rejected a bill to legalize same-sex marriage in 2009. Wednesday's vote was the fourth time same-sex marriages were approved in the Assembly, according to O'Donnell.

Currently, 31 senators, including two Republicans, are in favor of the bill. Backers need one more GOP member to vote in favor for it to pass.

Although New York does not grant same-sex marriages, a 2008 appellate court ruling upheld the right of couples to have their same-sex marriages recognized if they are performed elsewhere.

The act would amend the state's Domestic Relations Law to say, among other changes, that "no application for a marriage license shall be denied on the ground that the parties are of the same or a different sex."

Five states -- Massachusetts, Connecticut, Iowa, Vermont and New Hampshire -- and the District of Columbia currently grant same-sex marriage licenses.

The issue also is being debated in California, where a federal judge ruled Tuesday that another judge who ruled against a ban on same-sex marriage and later revealed that he is gay showed no evidence he was prejudiced in the case. Supporters of the ban said they would appeal.

CNN's Laura Dolan and Leigh Remizowski contributed to this report.