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 (Deegan et al. Nature 1997, 389, 827) 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.