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 ages, fundamental questions remain over the response of the Antarctic ice sheets to orbital cycles. Furthermore, an understanding of the behaviour of the marine-based West Antarctic ice sheet (WAIS) during the `warmer-than-present' early-Pliocene epoch (~5-3Myr ago) is needed to better constrain the possible range of ice-sheet behaviour in the context of future global warming. Here we present a marine glacial record from the upper 600m 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 today and atmospheric CO2 concentration was as high as ~400p.p.m.v. (refs 5, 6). The evidence is consistent with a new ice-sheet/ice-shelf model that simulates fluctuations in Antarctic ice volume of up to +7m in equivalent sea level associated with the loss of the WAIS and up to +3m 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 melt under conditions of elevated CO2.