Senators on Wednesday approved a plan to double the number of officers along the U.S.-Mexico border, a key concession to Republicans who plan to join with Democrats in supporting a comprehensive immigration measure this week.

At the same time, House Republicans signaled that there will be no quick resolution to the months-long debate over the nation’s immigration laws, regardless of what happens in the Senate.

House Speaker John A. Boehner (R-Ohio) told colleagues Wednesday that he will only hold votes on immigration proposals that are supported by a majority of his own caucus, and another senior Republican lawmaker suggested that Congress might not settle the issue until next year at the earliest.

By a vote of 69 to 29, senators amended the immigration bill to include provisions that would double the size of the U.S. Border Patrol along the Mexican border, require the construction of 700 miles of fencing along the southern border and authorize the use of new radar and unmanned aerial drones to track illegal border crossings.

“Americans want immigration reform, but they want border security first,” said Sen. John Hoeven (R-N.D.), a key backer of the amendment who helped secure 15 GOP votes for it.

But a majority of the Senate Republican conference banded together against the proposal, calling it a costly and ineffective way to keep people from crossing into the United States illegally.

The amendment “throws money at the border, but it doesn’t get the job done,” said Sen. Charles E. Grassley (R-Iowa).

Supporters of the comprehensive immigration bill, led by the bipartisan “Gang of Eight,” hope to hold a final vote by Thursday evening. But Senate Majority Leader Harry M. Reid (D-Nev.) complained again Wednesday that talks on wrapping up debate had gone “backwards” amid GOP objections to proceeding.

One potential sticking point was settled Wednesday when the U.S. Supreme Court struck down a key part of the federal Defense of Marriage Act, which denies federal benefits to same-sex couples who are legally married.

In response, Sen. Patrick J. Leahy (D-Vt.) withdrew two proposed amendments to the immigration bill that would have allowed same-sex foreign spouses and partners of U.S. citizens to apply for visas. Senate Republicans had threatened to jettison the entire package if the same-sex provisions were included.

Boehner, meanwhile, once again told his colleagues at their weekly closed-door meeting that no matter what the Senate does on immigration reform, the House will act on its own.

“It’ll be a bill that reflects the will of our majority and the people we represent,” Boehner said, according to GOP aides in the room.

Boehner’s reassurances pleased rank-and-file members, most of whom want the chamber to deal with border security before addressing the legal status of the nation’s estimated 11 million illegal immigrants.

“I think this train is getting ready to slow down” in the House, said Rep. Tom Cole (R-Okla.), a key Boehner ally.

Cole predicted that the House won’t seriously debate any immigration legislation until congressional leaders and the White House broker deals to fund government operations in the fiscal year that begins in October and to raise the federal debt ceiling.

“I don’t see how a big immigration brouhaha helps you going into trying to figure out how to keep the government funded and what to do about the debt ceiling,” Cole said.

Rep. Raul Labrador (R-Idaho), a tea party-backed lawmaker and immigration lawyer, said the House should move methodically and should not accelerate the process to seek political gains with the nation’s fast-growing Hispanic population.

“We’re running around like chickens with our heads cut off, thinking that we have to do this for political reasons,” Labrador told reporters. “We don’t have to do this for political reasons. In fact, the biggest mistake we can make as conservatives is to pander to the Hispanic community.”

Rep. Matt Salmon (R-Ariz.), another outspoken conservative, agreed. “If we’re going to do it for political expedience, it’s the flat-out wrong reason to do anything,” he said. “Everything we do should be based on sound, right policy.”