We show here both experimentally and theoretically that the formation of "coffee-ring" deposits observed at the edge of drying water droplets requires not only a pinned contact line but also suppression of Marangoni flow. For simple organic fluids, deposition actually occurs preferentially at the center of the droplet, due to a recirculatory flow driven by surface-tension gradients produced by the latent heat of evaporation. The manipulation of this Marangoni flow in a drying droplet should allow one in principle to control and redirect evaporation-driven deposition and assembly of colloids and other materials.