House Republicans appeared eager for that fight. For two years, the House has passed detailed but nonbinding budget plans that would cut domestic programs to levels not seen since World War II, enact changes to Medicare that would offer older people fixed subsidies to buy private health insurance, and mandate a much-simplified tax code. Democrats have criticized those plans, declined to produce an alternative, and instead demanded what they called a “balanced approach” to deficit reduction.

Now, Republicans said, the debate will be over numbers.

“We have a budget that’s described as draconian, that decimates this program or that. They have a phrase, ‘balanced approach,’ ” said Representative Trey Gowdy, Republican of South Carolina. “I’m tired of debating against a phrase.”

The debt ceiling legislation — devised with awareness of the constitutional hurdles imposed by the 27th Amendment on Congressional pay — would impound lawmaker salaries until a budget is passed or the 113th Congress ends, whichever comes first. And it would not require the House and the Senate to come to a compromise on the two spending and tax blueprints, which are likely to be very different. That will be the really difficult task.

House Democratic leaders tried to persuade their members to vote against the deal, so as to force as many Republicans as possible to vote to do something most said they would never do: lift the debt ceiling. But 86 Democrats voted yes, more than enough to let 33 Republicans vote no without bringing the bill down and handing Republican leaders an embarrassing defeat.

House Republicans say punting the debt ceiling to May 18 is not so much a retreat as a “re-sequencing” of the coming budget showdowns. House Republicans now take for granted that the first deadline, March 1, will come and go, and $110 billion in across-the-board spending cuts to military and domestic programs — known as a sequester — will go into force.

“The sequester is going to go into effect on March 1 unless there are cuts and reforms that get us on a plan to balance the budget over the next 10 years. It’s as simple as that,” Mr. Boehner said.