The instability and accelerated melting of the Antarctic Ice Sheet are among the foremost elements of contemporary global climate change. The increased freshwater output from Antarctica is important in determining sea level rise, the fate of Antarctic sea ice and its effect on the Earth's albedo, ongoing changes in global deep-ocean ventilation, and the evolution of Southern Ocean ecosystems and carbon cycling. A key uncertainty in assessing and predicting the impacts of Antarctic Ice Sheet melting concerns the vertical distribution of the exported meltwater. This is usually represented by climate-scale models as a near-surface freshwater input to the ocean, yet measurements around Antarctica reveal the meltwater to be concentrated at deeper levels. Here we use observations of the turbulent properties of the meltwater outflows from beneath a rapidly melting Antarctic ice shelf to identify the mechanism responsible for the depth of the meltwater. We show that the initial ascent of the meltwater outflow from the ice shelf cavity triggers a centrifugal overturning instability that grows by extracting kinetic energy from the lateral shear of the background oceanic flow. The instability promotes vigorous lateral export, rapid dilution by turbulent mixing, and finally settling of meltwater at depth. We use an idealized ocean circulation model to show that this mechanism is relevant to a broad spectrum of Antarctic ice shelves. Our findings demonstrate that the mechanism producing meltwater at depth is a dynamically robust feature of Antarctic melting that should be incorporated into climate-scale models.