Thirty years after oxygen isotope records from microfossils deposited in ocean sediments confirmed the hypothesis that variations in the Earth's orbital geometry control the ice ages¹, fundamental questions remain over the response of the Antarctic ice sheets to orbital cycles². Furthermore, an understanding of the behaviour of the marine-based West Antarctic ice sheet (WAIS) during the 'warmer-than-present' early-Pliocene epoch (5–3 Myr ago) is needed to better constrain the possible range of ice-sheet behaviour in the context of future global warming³. Here we present a marine glacial record from the upper 600 m of the AND-1B sediment core recovered from beneath the northwest part of the Ross ice shelf by the ANDRILL programme and demonstrate well-dated, 40-kyr cyclic variations in ice-sheet extent linked to cycles in insolation influenced by changes in the Earth's axial tilt (obliquity) during the Pliocene. Our data provide direct evidence for orbitally induced oscillations in the WAIS, which periodically collapsed, resulting in a switch from grounded ice, or ice shelves, to open waters in the Ross embayment when planetary temperatures were up to 3 °C warmer than today⁴ and atmospheric CO₂ concentration was as high as 400 p.p.m.v. (refs 5, 6). The evidence is consistent with a new ice-sheet/ice-shelf model⁷ that simulates fluctuations in Antarctic ice volume of up to +7 m in equivalent sea level associated with the loss of the WAIS and up to +3 m in equivalent sea level from the East Antarctic ice sheet, in response to ocean-induced melting paced by obliquity. During interglacial times, diatomaceous sediments indicate high surface-water productivity, minimal summer sea ice and air temperatures above freezing, suggesting an additional influence of surface melt⁸ under conditions of elevated CO₂.