Injection of carbon dioxide (CO2) into saline aquifers is a promising tool for reducing anthropogenic CO2 emissions. At reservoir conditions, the injected CO2 is buoyant relative to the ambient groundwater. The buoyant plume of CO2 rises toward the top of the aquifer and spreads laterally as a gravity current, presenting the risk of leakage into shallower formations via a fracture or fault. In contrast, the mixture that forms as the CO2 dissolves into the ambient water is denser than the water and sinks, driving a convective process that enhances CO2 dissolution and promotes stable long-term storage. Motivated by this problem, we study convective dissolution from a buoyant gravity current as it spreads along the top of a vertically confined, horizontal aquifer. We conduct laboratory experiments with analog fluids (water and a mixture of methanol and ethylene glycol) and compare the experimental results with simple theoretical models. Since the aquifer has a finite thickness, dissolved buoyant fluid accumulates along the bottom of the aquifer, and this mixture spreads laterally as a dense gravity current. When dissolved buoyant fluid accumulates slowly, our experiments show that the spreading of the buoyant current is characterized by a parabola-like advance and retreat of its leading edge. When dissolved buoyant fluid accumulates quickly, the retreat of the leading edge slows as further dissolution is controlled by the slumping of the dense gravity current. We show that simple theoretical models predict this behavior in both limits, where the accumulation of dissolved buoyant fluid is either negligible or dominant. Finally, we apply one of these models to a plume of CO2 in a saline aquifer. We show that the accumulation of dissolved CO2 in the water can increase the maximum extent of the CO2 plume by several fold and the lifetime of the CO2 plume by several orders of magnitude.