Gov. David A. Paterson plans to introduce legislation on Thursday to make marriage between same-sex couples legal in New York, advancing his push for greater rights for gay men and lesbians, at a time when other states have done so.

Mr. Paterson’s plans represent the most public effort yet by the governor, who has been a consistent supporter of gay rights, to position himself and New York at the crest of a broadening national movement.

The move allows him to lead on an issue that could prove defining in his governorship, which has so far been marked by political missteps and the crumbling economy.

But it is also something of a political gamble, because the legislation faces an uphill climb in the State Senate. Democrats acknowledge that they do not have enough votes there to get the measure approved, meaning that its chances could rest in the hands of a few Republicans.