The Senate took another step Monday to pass a bill to repeal the Defense of Marriage Act and codify federal recognition of same-sex marriage, with a bipartisan procedural vote on a compromise substitute amendment that includes religious liberty protections.

The bill would repeal the 1996 law that the Supreme Court found to be largely unconstitutional in a 2013 decision. It would also codify federal recognition of same-sex marriages that are legal in the state where the marriage was performed.

Following Monday’s 61-35 vote Senate Majority Leader Charles E. Schumer teed up possible votes on Republican amendments on Tuesday that could precede a final passage vote.

In a floor speech Monday, Schumer, D-N.Y., said he hopes the chamber “can get it done with all due speed, because millions of Americans deserve equal justice under the law and the peace of mind knowing their right to marry the person they love is protected.”

Earlier this month, the Senate voted 62-37 to move forward on the bill after bipartisan negotiators announced the compromise language in the amendment. Following that vote, leaders had hoped to pass the bill before Thanksgiving but conservatives in the chamber did not agree to hold a swift vote.

Texas Sen. John Cornyn and other Republicans have voted against the bill, arguing same-sex marriages are already protected by a 2015 Supreme Court decision that guaranteed constitutional protections for same-sex marriage.

“While this bill does not move the needle on same-sex marriage, this bill will raise questions on religious liberty,” Cornyn said in a floor speech Monday.

Earlier this year, the House passed a broader bill that would mandate all states honor out-of-state marriages regardless of race, gender or sexual orientation of the couple.

House Majority Leader Steny Hoyer, D-Md., said in a message about lame-duck period priorities that the chamber plans to advance the Senate-passed version of the measure.

“The Senate will soon pass the Respect for Marriage Act, and we must be prepared to take another vote on that historic bill to send it to the White House,” Hoyer said.

The congressional push for the bill followed the Supreme Court decision earlier this year that wiped out a constitutional right to abortion and raised questions about the court’s approach to other previous rulings about rights.

In a concurring opinion, Justice Clarence Thomas said the approach that led the court to overturn Roe v. Wade should also apply to the decisions undergirding the rights to same-sex marriage, same-sex relationships and access to contraception.