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There is another version of that story that has her walking during the foxhunt and falling. She was rushed back to Blenheim in a pony carriage, and the bumpy ride brought on the labor pains.

There has been another popular story on Winston being born in a ladies' room at Blenheim after his mother went into labor while dancing at a ball there.

Take your pick from what snopes has deduced: http://www.snopes.com/history/world/churchill.asp