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Washington (CNN) -- The military's prohibition of openly gay people serving within its ranks is one step closer to ending, after the Senate voted Saturday to repeal the armed forces' "don't ask, don't tell" policy.

Eight Republicans and independent Joe Lieberman of Connecticut joined the chamber's Democrats to back the legislation, which passed around 3 p.m. by a 65-31 margin. The bill needed a simple majority -- meaning support from 51 of the Senate's 100 members -- to pass.

"I want to thank all of the gay men and women who are fighting for us today," said Sen. Susan Collins, a Maine Republican who pushed hard for the measure. "We honor your service, and now we can do so openly."

President Barack Obama will sign the bill into law next week, White House press secretary Robert Gibbs said in a Twitter post moments after the Senate vote.

"Gay and lesbian service members -- brave Americans who enable our freedoms -- will no longer have to hide who they are," Obama said in an e-mailed statement to supporters. "The fight for civil rights, a struggle that continues, will no longer include this one."

Defense Secretary Robert Gates said, in a statement, that he'll "approach this process deliberately, (making changes) only after careful consultation with the military service chiefs and our combatant commanders."

Clifford Stanley, a retired Marine and the Under Secretary of Defense for Personnel and Readiness, will "immediately proceed" to execute the Pentagon's plan, Gates said.

The entire process, Collins estimates, will take "months, not years" to complete. Until then, Gates said that "the current law and policy will remain in effect."

Still, Sen. Kirsten Gillibrand, D-New York, said Saturday afternoon that she believed no man or woman would be outed or kicked out of the military in the meantime.

Adm. Mike Mullen, chairman of the Joint Chiefs of Staff, applauded Congress's action. He and Gates were among the civilian and active-duty military leaders who had rallied for the change.

"It is the right thing to do," he said in a statement. "No longer will able men and women who want to serve and sacrifice for their country have to sacrifice their integrity to do so. We will be a better military as a result."

"Don't ask-don't tell" became law in 1993, after opposition ballooned to then newly elected President Bill Clinton's plan to lift the military's complete ban on gay service members. The policy stopped the practice of asking service members if they are gay, but still called for the dismissal of openly gay service members.

Secretary of State Hillary Clinton called Saturday's vote a "historic step forward ... toward a more perfect union and a more perfect reflection of our core values."

A Pentagon study released earlier this month concluded that allowing openly gay or lesbian troops to serve in the military would have little lasting impact on the U.S. armed forces. Opposition to the change was much higher in Army and Marine combat units than in the military as a whole.

Obama and fellow Democrats were pushing for a speedy repeal, before the more conservative incoming Congress is seated the first week of January. Gates warned that court challenges to "don't ask, don't tell" could force an abrupt repeal of the policy, rather than the process in the legislation that would allow the military to manage the change on a longer timetable.

The House of Representatives acted first, passing the measure -- by a 250 to 175 margin -- last Wednesday. Four days later, Speaker of the House Nancy Pelosi, D-California, cheered the Senate for following suit.

"Today's landmark vote closes the door on a fundamental unfairness," Pelosi said in a statement. "It reflects a core principle in our nation: that anyone who wishes to serve, secure and defend this country should be welcomed, judged by their abilities, and honored for their sacrifice."

Gay rights groups cheered the bill's passage, sending out congratulatory statements soon after a key cloture vote Saturday morning paved the way for the final tally in the afternoon. After that first vote, repeal supporters hugged and shook hands in the corridor off the Senate floor, and Lieberman said one of his aides reported "great jubiliation" at the White House as well.

Joe Solmonese, the president of the Human Rights Campaign -- a lesbian, gay, bisexual and transgender equal rights group -- said "America lived up to its highest ideals of freedom and equality" with the vote.

"Plenty of people had already planned the funeral for this legislation," he said. "Today, we pulled out a victory from what was almost certain defeat just a few days ago."

While Democrats roundly rallied behind the policy change, Republicans largely opposed it.

Sen. Lindsay Graham, R-South Carolina, cited a Marine Corps commandant as saying he believes that "changing this policy this way would cause distraction among the Marine Corps to the point that he is worried about increased casualties."

"Let's hope he's wrong," Graham said Saturday, "but you've got to ask yourself, is he crazy to say that?"

Ultimately, eight of Graham's Republican colleagues ended up supporting the repeal on Saturday. In addition to Collins, they were Sens. Scott Brown of Massachusetts, Olympia Snowe of Maine, Lisa Murkowski of Alaska, Mark Kirk of Illinois, Richard Burr of North Carolina, John Ensign of Nevada and George Voinovich of Ohio.

Voinovich said he made up his mind after the reviewing the Defense Department's report on the policy.

"I accept its findings and Secretary Gates' recommendation and reassurance that the repeal will be implemented when the battle effectiveness of our forces is assured and proper preparations have been completed," he said.

The executive director of Servicemembers United, the nation's largest organization of gay and lesbian troops, said after the key cloture vote Saturday that the legislation's passage marked a "historic step forward for this country" and "very likely be a life-changing moment for gay and lesbian troops."

That includes men such as Alexander Nicholson, a former multi-lingual Army interrogator discharged under the policy.

While noting a certification and "yet-to-be-determined implementation period" awaits, he said Saturday, "Those who defend our freedom while living in fear for their careers will finally breathe a sigh of relief tonight, and those who have fallen victim to this policy in years past will finally begin to see true closure and redemption on the horizon."

CNN's Ted Barrett contributed to this report.