Friday evening, one of the Republican senators thought to be open to voting in favor of same-sex marriage announced that he would oppose the bill. Senator Andrew J. Lanza, of Staten Island, who had been heavily lobbied by gay rights advocates and Mr. Cuomo, said he believed the word marriage should “describe a union between a man and a woman.”

“Calling a union between a man and a woman marriage, and one between a man and man a civil union, does not invalidate one or the other,” he said in a statement.

Mr. Lanza, who has publicly agonized over the marriage issue, has been concerned that a yes vote on gay marriage would endanger his re-electability in his relatively conservative district.

The Senate easily defeated a same-sex marriage measure in 2009, when the chamber was controlled by Democrats; improbably, it seemed to face better odds this year, when it is controlled by Republicans, because Mr. Cuomo has succeeded in persuading all but one Democrat to support the measure, and because a few Republicans have changed their votes. Mr. Cuomo, a first-term Democrat, has said that passage of same-sex marriage is one of his top priorities.

The decision by Senate Republican leaders to allow a vote followed a week of intense behind-the-scenes negotiation between Mr. Cuomo and Senate Republicans, much of it focused on whether or not the governor’s proposal went far enough in protecting religious institutions that do not approve of same-sex unions. On Friday, the legislative leadership reached an agreement on a measure that they said would protect those religious institutions; that measure was approved Friday evening in the Assembly.

Opponents of same-sex marriage rallied in force on Monday, gathering outside the Senate chambers to sing and chant, as their leaders delivered petitions with 63,000 signatures of New Yorkers opposing the legalization of same-sex weddings.

“Regardless of the issue, there’s a process that should be followed, and that process is being violated tonight,” the Rev. Jason J. McGuire, the executive director of New Yorkers for Constitutional Freedoms, which advocates against same-sex marriage, said Friday. “They should have allowed this language to be out for a time to properly vet it. We are just now looking at this language, and we don’t have time to respond to all the legal concerns.”