ST. PAUL  — Minnesota’s proposed constitutional ban on gay marriage is history. The fight isn’t over for gay marriage supporters, though. The next logical step is the actual right to marry.

Though the ban was defeated by voters Tuesday, state law still prohibits gay marriage. New Democratic majorities in the state House and Senate make it more likely that the 1997 “Defense of Marriage” law could be repealed. Democratic Gov. Mark Dayton, who would have to approve such a change, is in favor of gay marriage.

But Dayton and Democratic legislative leaders played down that possibility Wednesday, at least in the immediate future. The likely new Senate leader, Sen. Tom Bakk of Cook, said policy changes would have to take a back seat to the state budget when the Legislature convenes in January.

“I’m not going to get into this particular or that particular,” Dayton said, asked if he would sign a bill to legalize gay marriage.

Despite those cautious words, Democratic Sen. John Marty of Roseville said he’d start pushing a legalization bill as soon as the session starts.