When a suspension of small particles is overlain by a clear fluid whose density is greater than that of the interstitial fluid, but less than that of the bulk suspension, the settling of the dense suspended particles can lead to vigorous convection in the overlying fluid. This novel situation is investigated experimentally and theoretically. A sharp interface is observed between the convecting upper region and a stagnant lower region in which there is unimpeded sedimentation at low Reynolds number. There is no transport of fluid from the upper region into the lower, though there is mixing of both buoyant fluid and entrained particles from the lower region into the upper. The interface between the two regions is found to descend at a constant velocity. Systematic laboratory measurements have determined how this velocity depends on the densities of the layers and the distributions of settling velocities of the particles. A theoretical description is developed which calculates the evolution of the density of the lower region due to differential sedimentation of polydisperse particles. Buoyancy arguments based on the calculated density profile are used to place upper and lower bounds on the amount of particle entrainment into the upper layer and on the rate of fall of the interface between the convecting and sedimenting regions. The theoretical predictions are in good agreement with the experimental observations. The analysis of the interaction between convection and sedimentation in the system considered here may be particularly relevant to the description of evolving crystal-rich layers in magma chambers and of silt-laden outflow from rivers, and has a wide range of other industrial, environmental and geological applications.