All year long, House Speaker John Boehner has struggled to corral his large but raucous Republican majority. Now comes the ultimate test.

Mr. Boehner told his rank-and-file Sunday night that he had come to agreement with the White House on a deficit-cutting deal that still needs to pass through his chamber. It's less conservative than the one he jammed through last week, and the vote could go down-to-the-wire.

At...