An agreement to vastly increase fencing, patrols and high-tech monitoring along the U.S.-Mexico border was formally unveiled in the Senate on Friday, providing powerful momentum to a far-reaching immigration bill backed by the White House.

With the border security amendment finalized, the immigration legislation immediately picked up an additional likely Republican supporter: Sen. Lisa Murkowski of Alaska, who signed on as a co-sponsor of the amendment.

“This amendment will put to rest any remaining credible concerns about the border, about border security,” Senate Majority Leader Harry Reid, D-Nev., said on the Senate floor as he filed the measure and announced procedural steps to bring it to a vote early next week. “The opposition of a small group is not going to stop this bill from moving forward,” Reid said.

The deal doubling Border Patrol agents and adding hundreds of miles of fencing along the Southwest border had already won support from four other undecided Republican senators who are now likely to back the immigration bill when it comes to a final vote next week. The legislation opening the door to citizenship for millions now appears within reach of securing the broad bipartisan majority that its authors say is needed to ensure serious consideration by the GOP-controlled House.

However, the outcome there remains far from certain because many conservatives are opposed to offering citizenship to people who came to this country illegally.

“We really have tried to secure the border in a way that we hope can get bipartisan support and that Americans want,” Sen. John Hoeven, R-N.D., an author of the amendment, said in a phone interview Friday. “We’re hopeful to have a good bipartisan majority.”

Sen. John McCain, R-Ariz., said on Fox News Channel on Friday that “if there’s anyone who still will argue that the border is not secure after this, then border security is not their reason for opposing a path to citizenship for the people who are in this country illegally.”

“Is it more than I would have recommended? Honestly, yes,” McCain said. “But we’ve got to give people confidence.”

Hoeven developed the amendment along with Republican Sen. Bob Corker of Tennessee, in consultation with McCain, Sens. Lindsey Graham, R-S.C., Chuck Schumer, D-N.Y., and other members of the so-called Gang of Eight who wrote the immigration bill. It prevents immigrants now here illegally from attaining permanent resident status until a series of steps have been taken to secure the border.

These include doubling the Border Patrol with 20,000 new agents, 18 new unmanned surveillance drones, 350 miles of new pedestrian fencing to add to 350 miles already in place, and an array of fixed and mobile devices to maintain vigilance, including high-tech tools such as infrared ground sensors and airborne radar.

The new provisions would be put in place over a decade, in line with the 10-year path to a permanent resident green card that the bill sets out for immigrants here illegally. During that time, the immigrants could live and work legally in a provisional status.

Hoeven said the 10-year cost of the border security amendment included $25 billion for the additional Border Patrol agents, $3 billion for fencing and $3.2 billion for other measures.

White House press secretary Jay Carney said the border deal “would constitute a breakthrough” on immigration. “We’re pleased that Republicans and Democrats continue to work together toward comprehensive immigration reform,” he said.