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The Senate on Wednesday cleared a key procedural hurdle toward historic passage of the bipartisan bill to protect same-sex and interracial marriage, voting 62-37 to break a filibuster.

Wednesday’s successful test vote signals the bill is on a glide path to succeed, a remarkable turn of events given how contentious the issue of same-sex marriage was just a few years ago.

While the bill would not set a national requirement that all states must legalize same-sex marriage, it would require individual states to recognize another state’s legal marriage. So, in the event the Supreme Court might overturn its 2015 Obergefell v. Hodges decision that legalized same-sex marriage, a state could still pass a law to ban same-sex marriage, but that state would be required to recognize a same-sex marriage from another state.

All 50 members of the Democratic caucus voted to start debate on the bill as well as 12 Republicans.

Senate Majority Leader Chuck Schumer on Thursday set the next procedural vote on same-sex marriage bill for November 28 – after senators’ Thanksgiving recess. It will require 60 votes, which it’s expected to get, before a final passage vote in the chamber later that week.

Democrats couldn’t get an agreement with Republicans to expedite the floor process.

Schumer had told CNN he wanted his chamber’s bill to pass by Thursday before senators leave for their Thanksgiving recess all next week.

“We’re hoping that could happen,” he said.

Earlier this week, Schumer expressed “hope” that after the vote Wednesday, “both sides can work quickly together to move this bill through the Senate and on to the president’s desk.”

“It already passed the House earlier this year with significant 47 Republican votes and I’m optimistic we can achieve a significant result in this chamber,” he added.

Once the bill passes the Senate, it will need to be passed again through the House before going to President Joe Biden’s desk to be signed into law. Supporters of the bill hope to pass the legislation through the House before the end of the year as Republicans appear on track to take control of the chamber in the next Congress.

Earlier this week, the bipartisan negotiators who worked on the legislation, announced they were “confident” the bill has enough votes to pass and were hoping the bill could be put to the floor for a vote.

The bipartisan group, which includes Republican Sens. Rob Portman of Ohio, Susan Collins of Maine and Thom Tillis of North Carolina and Democratic Sens. Tammy Baldwin of Wisconsin and Kyrsten Sinema of Arizona, said in a statement Monday that they “look forward to this legislation coming to the floor.”

Lawmakers had hoped to pass the bill before leaving for recess ahead of the midterm elections, but the chamber punted on a vote until after the November elections as negotiators asked for more time to lock down support. That gamble appears to have paid off for the bill’s supporters given the 12 Republican votes to break the filibuster Wednesday.

In a sign of how much support has grown in recent years for same-sex marriage, the bill found backing from GOP senators including those in deeply red states.

Republican Sen. Cynthia Lummis of Wyoming told CNN’s Manu Raju that she voted to advance the Senate’s same-sex marriage bill due to “Article 1, Section 3 of the Wyoming Constitution,” which she read to reporters and includes an anti-discrimination clause.

“That’s why we’re called the equality state,” she added.

Utah Sen. Mitt Romney said the “bill made sense” and “provides important religious liberty protections.”

“While I believe in traditional marriage, Obergefell is and has been the law of the land upon which LGBTQ individuals have relied,” Romney said in a statement. “This legislation provides certainty to many LGBTQ Americans, and it signals that Congress—and I—esteem and love all of our fellow Americans equally.”

This story and headline have been updated to reflect additional developments.

CNN’s Manu Raju and Clare Foran contributed to this report.