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Russian leader Nikita Khrushchev has agreed to dismantle all Russian missiles based in Cuba and ship them back to the Soviet Union.
The announcement was made in a public message to President John F Kennedy broadcast on Moscow Radio.
In response, President Kennedy said the decision to remove the Cuban missiles was an "important contribution to peace".
He has also promised the US will not invade Cuba and will eventually lift the US naval blockade imposed on the island.
The blockade will continue until effective UN inspection ensures that the missiles in Cuba have been dismantled.
Spy plane uncovers missiles
The crisis began on 14 October after a U-2 reconnaissance plane revealed the existence of several nuclear missiles based in on the Caribbean island capable of reaching the US.
Then, a week ago, President Kennedy made a TV address to the nation denouncing the Soviets' actions. He declared a naval blockade on Cuba and threatened the USSR itself with attack if any Cuban missile were launched against the USA.
Since then the world has been on tenterhooks wondering which superpower would back down first, desperately hoping this was not the beginning of a nuclear World War III.
Four days ago, faced with a huge US fleet including eight aircraft carriers that had formed an arc 500 miles (800km) from the eastern tip of Cuba, Soviet vessels approaching the island turned back.
Relief and anger
The British Foreign Secretary the Earl of Home welcomed Khrushchev's decision to dismantle the weapons in Cuba and praised President Kennedy's diplomatic skills.
China has expressed anger at the Soviet climb-down and said it will support Cuba "through thick and thin".
As part of the settlement Cuba's president Fidel Castro, angered that he was not consulted on the agreement, has ordered all Americans off the American base at Guantanamo used by US military for 60 years.
In the summer of 1962 Nikita Khrushchev gave the go-ahead for nuclear missiles to be installed on Cuba to protect it from US invasion and also to counterbalance US superiority in long- and medium-range nuclear weapons based in Europe.
After a US spy plane spotted the missile bases, the news was announced by President Kennedy and for a week the world hovered on the brink of all-out nuclear war.
It was the Soviets who eventually backed down after Khrushchev insisted Kennedy promise not to invade Cuba. He also demanded the withdrawal of Jupiter missiles in Turkey.
Kennedy publicly agreed to the first request and secretly agreed to the second.
The US ended its blockade on 20 November 1962, the Soviets removed their nuclear weapons by the end of the year and US missiles in Turkey were withdrawn in 1963.
A hot line between the USA and USSR was set up to prevent such a crisis happening again.
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