Senate Republicans and Democrats on Sunday cemented a compromise plan to crack down on unlawful migration across the U.S. border with Mexico and cleared a critical hurdle to an aid package for Ukraine, but the deal faces long odds in a Congress deeply divided over both issues.

The release of the agreement, struck after more than three months of near-daily talks among senators and Biden administration officials, counted as an improbable breakthrough on a policy matter that has bedeviled presidents of both parties and defied efforts at compromise for decades on Capitol Hill. President Biden implored Congress late last month to pass it, promising to shut down the border immediately once it became law.

But Speaker Mike Johnson has pronounced it “dead on arrival” in the Republican-controlled House. And with former President Donald J. Trump actively campaigning against the deal, it was not clear whether the measure could even make it out of the Democratic-led Senate, where it needs bipartisan backing to move forward.

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