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Meltwater stored in ponds and crevasses can weaken and fracture ice shelves, triggering their rapid disintegration. This ice-shelf collapse results in an increased flux of ice from adjacent glaciers and ice streams, thereby raising sea level globally. However, surface rivers forming on ice shelves could potentially export stored meltwater and prevent its destructive effects. Here we present evidence for persistent active drainage networks-interconnected streams, ponds and rivers-on the Nansen Ice Shelf in Antarctica that export a large fraction of the ice shelf's meltwater into the ocean. We find that active drainage has exported water off the ice surface through waterfalls and dolines for more than a century. The surface river terminates in a 130-metre-wide waterfall that can export the entire annual surface melt over the course of seven days. During warmer melt seasons, these drainage networks adapt to changing environmental conditions by remaining active for longer and exporting more water. Similar networks are present on the ice shelf in front of Petermann Glacier, Greenland, but other systems, such as on the Larsen C and Amery Ice Shelves, retain surface water at present. The underlying reasons for export versus retention remain unclear. Nonetheless our results suggest that, in a future warming climate, surface rivers could export melt off the large ice shelves surrounding Antarctica-contrary to present Antarctic ice-sheet models, which assume that meltwater is stored on the ice surface where it triggers ice-shelf disintegration.