But Republicans who support immigration reform still face the task of winning more of their colleagues' votes. The Cornyn amendment has been presented by leaders as the measure that could assuage the fears of the bill's critics.
Senate Minority Leader Mitch McConnell (R-Ky.) on Tuesday called it "the key amendment" to winning the trust of most Republicans. Sen. Jeff Flake (R-Ariz.), a Gang of Eight member, said that Cornyn is sincere in his willingness to back the bill if his amendment is adopted.
"I can tell you, he wants to strengthen the bill," he said at a breakfast sponsored by the Christian Science Monitor. "And he has said himself he would like to vote for the bill."
But immigrant-rights advocates aren't so sure. Here's what Frank Sharry, executive director of America's Voice, told Fusion last week.
"He pretends to be sincere about the need for reform," he said. "He asks for changes that are a bridge too far. He destabilizes the bipartisan agreement already in place. He helps to thwart reform."
This post was updated at 6:31 PM