The bed of the West Antarctic Ice Sheet is, in places, more than 1.5 km below sea level(1,2). It has been suggested that a positive ice-loss feedback may occur when an ice sheet's grounding line retreats across a deepening bed(1-3). Applied to the West Antarctic Ice Sheet, this process could potentially raise global sea level(4) by more than 3 m. Hitherto, attention has focussed on changes at the Siple Coast(5-7) and Amundsen Sea embayment(8-10) sectors of West Antarctica. Here, we present radio-echo sounding information from the ice sheet's third sector, the Weddell Sea embayment, that reveals a large subglacial basin immediately upstream of the grounding line. The reverse bed slope is steep, with about 400 m of decline over 40 km. The basin floor is smooth and flat, with little small-scale topography that would delay retreat, indicating that it has been covered with marine sediment(5,11) and was previously deglaciated. Upstream of the basin, well-defined glacially carved fjords with bars at their mouths testify to the position of a former ice margin about 200 km inland from the present margin. Evidence so far suggests that the Weddell Sea sector of the West Antarctic Ice Sheet has been stable, but in the light of our data we propose that the region could be near a physical threshold of substantial change.