When carbon dioxide (CO2) dissolves into water, the density of water increases. This seemingly insubstantial phenomenon has profound implications for geologic carbon sequestration. Here we show, by means of laboratory experiments with analog fluids, that the up-slope migration of a buoyant current of CO2 is arrested by the convective dissolution that ensues from a fingering instability at the moving CO2-groundwater interface. We consider the effectiveness of convective dissolution as a large-scale trapping mechanism in sloping aquifers, and we show that a small amount of slope is beneficial compared to the horizontal case. We study the development and coarsening of the fingering instability along the migrating current and predict the maximum migration distance of the current with a simple sharp-interface model. We show that convective dissolution exerts a powerful control on CO2 plume dynamics and, as a result, on the potential of geologic carbon sequestration.