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Bootstrap percolation is a simple cellular automaton model. Sites in an L by L square are initially independently occupied with probability p. At each time step, an unoccupied site becomes occupied if it has at least two occupied neighbors. We study the behavior as p tends to 0 and L tends to infinity simultaneously of the probability I(L,p) that the entire square is eventually occupied. We prove that I tends to 1 if liminf p log L > lambda, and I tends to 0 if limsup p log L < lambda, where lambda=pi^2/ 18. The existence of lambda settles a conjecture of Aizenman and Lebowitz, while the determination of the value corrects numerical prediction of Adler, Stauffer and Aharony.