Indirect evidence suggests that volcanic activity occurring beneath the West Antarctic ice sheet influences ice flow and sheet stability(1-3). However, only volcanoes that protrude through the ice sheet(4) and those inferred from geophysical techniques(1,2) have been mapped so far. Here we analyse radar data from the Hudson Mountains, West Antarctica(5), that contain reflections from within the ice that had previously been interpreted erroneously as the ice-sheet bed. We show that the reflections are present within an elliptical area of about 23,000km(2) that contains tephra from an explosive volcanic eruption. The tephra layer is thickest at a subglacial topographic high, which we term the Hudson Mountains Subglacial Volcano. The layer depth dates the eruption at 207 BC +/- 240 years, which matches exceptionally strong but previously unattributed conductivity signals in nearby ice cores. The layer contains 0.019 - 0.31 km(3) of tephra, which implies a volcanic explosive index of 3-4. Production and episodic release of water from the volcano probably affected ice flow at the time of the eruption. Ongoing volcanic heat production may have implications for contemporary ice dynamics in this glacial system.