New York’s state Assembly was set to vote Wednesday night on a same-sex marriage bill, but the real battle is expected to be in the Senate.

The Marriage Equality Act 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 a news release from Gov. Andrew Cuomo’s office.

Cuomo said the state has “been on the right side of history,” citing such civil rights battles as women’s suffrage. “But on the issue of marriage equality, our state has fallen behind,” he said.

Democratic state Rep. Daniel O’Donnell said the bill has support in the state House.

Senate Republicans met Wednesday on the issue, and the timing of a vote there was unclear, according to CNN Albany affiliate WXAA.

The state Senate rejected a bill to legalize same-sex marriage in 2009.

“We know that the vast majority of New Yorkers is on our side, and we’ll keep struggling until the work is done,” said Ross Levi of Empire State Pride Agenda, according to WXAA.

Cuomo has been a vocal proponent of allowing same-sex couples in New York to marry. In March, he released a statement saying that he was in the midst of a series of meetings to discuss such a marriage equality bill.

“Marriage equality is a matter of fairness and legal security for thousands of families in this state — not of religion or culture,” Cuomo said Tuesday.

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.

CNN’s Laura Dolan contributed to this report.