ALBANY  Fourteen months after the State Senate soundly rejected legislation that would have legalized same-sex marriage, Gov. Andrew M. Cuomo is preparing to pursue another vote on the matter in the next few months.

Mr. Cuomo, who included legalizing same-sex marriage in his campaign platform, said Wednesday that he intended to ask the Legislature to take up the matter in its current session, which ends in June.

“We’ll be working very hard to pass it,” the governor, a Democrat, told reporters after delivering an encore of his budget address at Hofstra University on Long Island.

His pledge was greeted warmly by gay-rights activists, who have waited with some uncertainty to see whether Mr. Cuomo, faced with a daunting battle over the budget, would make a charged social issue like legalizing same-sex marriage a priority.