In this paper, we consider the fractional Laplacian -(-?)a/2 on an open subset in Rd with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such Dirichlet fractional Laplacian in C1.1 open sets. This heat kernel is also the transition density of a rotationally symmetric a-stable process killed upon leaving a C1.1 open set. Our results are the first sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets.