Don’t look now, but much of the National Security Agency bulk metadata collection that stirred so much controversy in the wake of the Edwards Snowden revelations might — just might — be about to come to an end. While civil libertarians still worry about various aspects of the program continuing, this would be no small achievement.

Yesterday a bipartisan group of Senators — led by Republican Mike Lee and Democrat Patrick Leahy — introduced the U.S.A. Freedom Act, a measure that would put an end to the NSA’s bulk collection under Section 215 of the Patriot Act. Things could go wrong from here on out, but it’s a possibility that something like this is going to end up becoming law soon enough — meaning the left-right alliance that has come together against bulk surveillance just might win a partial victory.

The reason the U.S.A. Freedom Act just might pass is because the alternative is unclear. On June 1st, Section 215 of the Patriot Act will sunset, so if nothing happens, the program will end entirely. Lawmakers are unlikely to allow that to happen.

To be sure, Senate “hawks” will try to prevent the U.S.A. Freedom Act from getting through. Mitch McConnell is pushing a bill that would reauthorize the NSA’s authority to conduct bulk surveillance until the end of the republic — sorry, until 2020. Republican aides who want reform expect McConnell to put up Senator Tom Cotton — a neocon who led the charge on that letter to Iran — as the front man to stop the U.S.A. Freedom Act.

But there’s a decent chance there will be 60 votes there to pass the U.S.A. Freedom Act in the end. That’s because it’s hard to imagine that there are enough votes in the Senate to pass clean re-authorization to keep bulk surveillance going. Very few Democrats will support that. And libertarian-leaning GOP Senators who are running for president are adamantly opposed to clean re-authorization: Ted Cruz supports the U.S.A. Freedom Act, and Rand Paul may want to go further.

If it does become clear that clean re-authorization can’t pass, GOP aides believe, the alternatives will then be to pass nothing — which causes all of Section 215 to sunset — or pass some version of the U.S.A. Freedom Act. Its list of co-sponsors is pretty strong: It includes Lee and Cruz and Leahy, and Republicans Dean Heller and Steve Daines, and Democrats Dick Durbin and Chuck Schumer. A companion version of the bill has been introduced in the House, where we’ve already seen that there is a left-right alliance against bulk surveillance. A good showing in the House — and this bill is supported by Republicans like James Sensenbrenner and Bob Goodlatte — might help get this passed in the Senate.

If so, Obama would probably sign it. The Obama administration has signaled support, as this statement from National Security Council spokesman Ned Price indicates:

We look forward to partnering with leaders of both parties in the House and the Senate as they work to pass a bill ahead of the June expiration of important intelligence authorities. The introduction of this legislation is an important step in the right direction.

After the Snowden revelations prompted a national outcry, Obama said he no longer wants the NSA to hold phone records in bulk, but left the details to Congress. The U.S.A. Freedom Act would end such bulk surveillance, require the U.S.’s Foreign Intelligence Surveillance Court to release redacted versions of its opinions in individual cases where the government is seeking data, and create a public privacy advocate to counter the government before the court. As Senator Durbin puts it, the bill is designed to “strike a balance” between making “critical reforms to bulk collection of Americans’ telephone and internet records” while “protecting America’s national security.”

To be sure, civil libertarians don’t think the bill goes far enough. Neema Singh Guliani, the legislative counsel for the ACLU, argues that the government still retains the authority to conduct mass surveillance under other authorities outside Section 215. “Whether Congress enacts the USA Freedom Act or lets Section 215 expire, they will need to put in place more sweeping reform,” she says, adding that the bill “still leaves the door open to broad government surveillance that sweeps up the communications of people who are not associated with terrorism.”

Still, Guliani notes, “the bill would effectively end nationwide collection under some authorities.”

If this does become law, which seems very possible, it would be an achievement for the left-right alliance that has been battling NSA bulk surveillance for years now — particularly at a time when ISIS and the possibility of a nuclear deal with Iran, combined with the GOP presidential primary, have the hawks on something of a rampage.