It was recovered from a meeting room in Hamilton, Bermuda by a soldier of the First Battalion Royal Welsh Fusiliers who was tasked with guarding the room during the talks.
Churchill (1874-1965) had been meeting with President Eisenhower of the US and Premier Laniel of France.
The three leaders met to discuss issues of the day - including West Germany's integration into the rest of Europe, the Korean war and the rising tensions over the Suez canal.
One soldier present at the conference, Brian Mead, commented on Churchill's appearance at the time in a recent piece for Living Magazines Cardiff: "He came through the entrance supported on either arm by two men.
"An exhausted little old fat man who was obviously partially paralysed from head to foot on his left side.
"He shuffled, rather than walked, and after murmuring greetings to those introduced to him, he was quietly assisted to the lift, followed by an embarrassing silence as we watched him leave."
The prime minister's ill health was due to a major stroke he had suffered earlier in the year. It would not be revealed to the press until after the talks took place.
The brand is La Aroma de Cuba, one of two brands Churchill smoked.
Churchill cigars are strong performers at auction. A half-smoked example from 1941 sold for ?�4,500 ($7,350) in 2010.
The auction ends on October 1.
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