The U.S. House on Thursday approved legislation requiring documentary proof of U.S. citizenship for anyone registering to vote, something voting rights groups have warned could disenfranchise millions of Americans.

The bill, known as the Safeguard American Voter Eligibility Act, or the SAVE Act, now heads to the Senate, where its fate is uncertain because Republicans don’t have a large enough majority to avoid a filibuster.

If it eventually becomes the law, the SAVE Act would take effect immediately and apply to all voter registration applications.

“This has no impact on individuals that are currently registered to vote,” said Rep. Bryan Steil, a Wisconsin Republican who has been advocating for the bill.

Voting rights groups say there is more to the story.

The law would affect voters who already are registered if they move, change their name or otherwise need to update their registration. That was acknowledged to some extent by the bill’s author, Republican Rep. Chip Roy of Texas, during a recent hearing on the legislation.

▶ Read more about how the bill could affect voting in the U.S.