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The government agreed to the payment of a ransom of $53 million in food and medical supplies, donated by companies all over the USA, as a condition for their release.
The airlift of the prisoners began yesterday, when the first 107 men boarded a DC6 airliner supplied by Pan American World Airways at a military airbase near Havana.
After just four flights, however, the operation was suspended for the night, to the consternation of the thousands of anxious relatives of the prisoners, keeping vigil in Florida for their return.
Flights resumed early this morning, and by the end of the day all 1,113 prisoners had been safely returned.
A rapturous crowd of 10,000 Cuban exiles greeted each new arrival at the Dinner Key Auditorium, on the outskirts of Miami.
Police and army cordons could barely restrain the mass of cheering, flag-waving people as they surged towards the buses bringing the prisoners back.
The release was the result of almost a year of talks between the Cuban government and a New York lawyer, James B Donovan, acting on behalf of the relatives of the prisoners.
The negotiations had the covert backing of the US government.
Mr Donovan persuaded the Cuban leader, Fidel Castro, to accept supplies of medicine and food instead of cash in exchange for the prisoners.
Mr Castro has demanded that a fifth of it should be delivered before Christmas Day.
Delivery 'could take months'
The first down payment was delivered yesterday by the 6,000-ton American freighter, the African Pilot.
Full delivery of the entire list of 10,000 items, all donated by companies and organisations across the United States, is expected to take months.
The prisoners were taken after the disastrous Bay of Pigs invasion of Cuba on 17 April 1961, funded in part by the US government.
Fidel Castro's militia crushed the invasion within a few days and captured 1,200 of the 1,500 invaders.
They were sentenced to up to 30 years in prison at their trial in Havana earlier this year.
Passing sentence, the court added that a ransom of $62m would secure their release.
Cuban exiles in the United States arranged the return of 60 wounded prisoners soon afterwards for $2.5m (about £900,000).
The remainder have been in captivity ever since.
It was later revealed that the ransom deal almost fell apart at the last minute, after Fidel Castro demanded an additional $2.9m in cash just as the prisoners were preparing to leave.
The money was raised in a frantic day of fundraising on 24 December by the president's brother, Attorney General Robert F. Kennedy, and General Lucius D Clay, an advisor to the Cuban Families Committee of prisoners' relatives.
The largest donation was $1m, given by a donor who wished to remain anonymous but was not, the government said, part of the Kennedy family.
The prisoners' release came at the end of an all-time low in relations between the United States and Cuba.
Diplomatic relations, broken off in 1960 after Fidel Castro nationalised US businesses in Cuba, have never been restored, and the US continues to enforce trade sanctions against Fidel Castro's regime.
After the collapse of the Soviet Union, Cuba was forced to introduce limited market reforms.
But refugees continue to make the perilous 90-mile journey to join the growing community of Cuban exiles in Florida.
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