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Clay, from Kentucky, was announced the winner after the hot favourite retired at the end of the sixth round in Miami.
When the bell rang for the start of the seventh round, Liston stayed on his stool in the corner of the ring - saying he did not want to continue.
Florida state attorney Richard Gerstein is to launch an inquiry into the fight.
Clay was earlier fined around £900 for disgraceful conduct after he ranted at his 32-year-old opponent during the weigh-in.
He had chanted "I wanna rumble...I wanna rumble!" before saying "You're a tramp. I am going to eat you up. Somebody's going to die at the ringside tonight. Are you scared?"
Liston had built his reputation on two first round knock-outs of Floyd Patterson and in the boxing world he was thought to be invincible.
Clay had come into the fight after suffering his greatest humiliation to date - being knocked down by Henry Cooper's left hook.
Liston reigned supreme when he was challenged by former Olympic gold medallist Clay, who ring experts said was "all talk".
The majority of sports journalists covering the bout thought that Clay, who was 7-1 underdog, would be no match for Liston. Only half of the seats had been sold and around 8,000 people attended the fight.
The fight started at 0310 GMT at the Convention Hall in Miami Beach and Liston had been widely expected to retain his title in the third round.
Liston, known as the "dark destroyer", had managed to pin Clay to the ropes during round two but in round three the champion was forced to retreat, with his left eye pouring with blood.
The World Boxing Association let Cassius Clay keep his heavyweight championship following the controversy over his fight with Sonny Liston at Miami Beach.
Ed Lassman, president of the WBA, had earlier demanded that Clay be stripped of his title and recommended giving the title back to him if his conduct improved over the next five months.
Miami State attorney Richard Gerstein said a month-long inquiry revealed no evidence of match-fixing. He added that there was little doubt that Liston was suffering from a sore shoulder during the fight.
Both the original fight and the rematch a year later in Maine, which Clay won with a first-round knockout, have been dogged by allegations of rigging.
Muhammad Ali, as he was later known to his adoring fans, said the victory in Miami over Liston was the most important of his career - because he had proved he was qualified to be champion of the world.
Shortly after the fight, Clay surprised the sports world by announcing that he had joined the Nation of Islam and had changed his name to Muhammad Ali.
His boxing career lasted 20 years, during which he won 56 fights and scored 37 knock-outs.
As an admirer once said, Ali would, "float like a butterfly, sting like a bee". In December 1981, Ali decided to retire from the ring and Parkinson's disease was later diagnosed.
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