We consider the nature of the fluid-solid phase transition in a polydisperse mixture of hard spheres. For a sufficiently polydisperse mixture (sigma>0.085) crystallization occurs with simultaneous fractionation. At the fluid-solid boundary, a broad fluid diameter distribution is split into a number of narrower fractions, each of which then crystallize. The number of crystalline phases increases with the overall level of polydispersity. At high densities, freezing is followed by a sequence of demixing transitions in the polydisperse crystal.