No. Although he grew up to be one of the most destructive dictators in history, Josef Stalin was physically normal as a baby. Also note, Adolf Hitler was not born with a moustache either.
He was born on 18 December 1878 and died 5 March 1953.
Joseph Stalin was born December 18th, 1878 in Gori, Georgia, Russia. He became the General Secretary of the Communist Party upon the death of Vladimir Lenin.
Georgia (called Sakartvelo by its inhabitants). Stalin was born in the town of Gori and went on to control the Soviet Union. He killed more people than Hitler and pretty much got away with it.
Josef Hassid was born in 1923.
Josef Gerstmann was born in 1887.
Josef Skružný was born in 1871.
Josef Lenz was born in 1934.
Josef Levi was born in 1938.
Josef Ternström was born in 1888.
Josef Walcher was born in 1954.
Josef Pieprzyk was born in 1949.
Josef Deckert was born in 1843.