The former Defence Department contract workers, Marc Gonsalves, Keith Stansell and Thomas Howes, returned to the United States on Wednesday evening.
U.S. President George W. Bush spoke by telephone with Uribe and praised the rescue operation, the White House said.
In Paris, French President Nicolas Sarkozy said, "Today a nightmare of more than six years has ended." Sarkozy, who had had made vigorous efforts to seek Betancourt's freedom, sent his foreign minister, Bernard Kouchner, to Colombia.
"I am filled with happiness," Betancourt's sister, Astrid, told Colombian radio. "These have been long years of waiting."
Betancourt was kidnapped by the FARC while campaigning for the presidency in 2002 when, against the advice of the armed forces, she travelled along a rural road in southern Colombia and was stopped at a rebel roadblock.
The presidents of Chile, Brazil and Peru praised Uribe for the rescue operation as a gain for peace and democracy.
Venezuelan President Hugo Chavez, a self-styled socialist revolutionary who has been at odds with Uribe over his support for the rebels, called the Colombian leader to congratulate him on the successful operation, Venezuelan state television said.
Chavez this year brokered the release of a group of hostages held by the FARC. But a Colombian army mission to kill a top FARC commander inside Ecuador triggered an Andean crisis that threatened to spill over into border violence.
Chavez, who had once called for more political recognition for the Marxist-inspired FARC, last month urged rebel commanders to release their hostages without conditions.
CONDITIONS FOR TALKS
The FARC has demanded that Uribe pull back troops from an area the size of New York City to facilitate talks.
Uribe, whose father was killed in a botched FARC kidnapping years ago, refuses to accept that condition. But he has offered a smaller safe haven under international observation in an area where there are no armed forces or armed groups.
The rescue weakens the FARC's position to negotiate as their ranks are thinned by military setbacks and desertions. But they hold scores more hostages for political leverage.
The outlawed rebel army, once a 17,000-member force able to attack cities and kidnap almost at will, has been driven back into remote areas and now has about 9,000 combatants. The guerrillas have lost three major leaders this year.
Listed as a terrorist group by U.S. and European officials, the FARC has used the cocaine trade to fund its operations.
Michael Shifter, an analyst at the Inter-American Dialogue, in Washington, said the rescue showed the FARC was in a serious organizational crisis.
"The Colombian government took advantage of the FARC's weakness and disarray to carry out the mission," he said. "It was a big gamble, but it worked."
