Silicon is the most important semiconducting material in the microelectronics industry.If current miniaturization trends continue, minimum device features will soon approach the size of atomic clusters. In this size regime, the structure and properties of materials often differ dramatically from those of the bulk. An enormous effort has been devoted to determining the structures of free silicon clusters [1-22]. Although progress has been made for Sin with n < 8, theoretical predictions for larger clusters are contradictory [2-22] and none enjoy any compelling experimental support. Here we report geometries calculated for medium-sized silicon clusters using an unbiased global search with a genetic algorithm. Ion mobilities  determined for these geometries by trajectory calculations are in excellent agreement with the values that we measure experimentally. The cluster geometries that we obtain do not correspond to fragments of the bulk. For n = 12-18 they are built on a structural motif consisting of a stack of Si9 tricapped trigonal prisms. For n >or= to 19, our calculations predict that near-spherical cage structures become the most stable. The transition to these more spherical geometries occurs in the measured mobilities for slightly larger clusters than in the calculations, possibly because of entropic effects.