The mostly terrestrial East Antarctic ice sheet is ten times larger and probably more stable than the mostly marine West Antarctic ice sheet. It is natural to suppose that the former appeared first, and that perhaps the latter was partially formed from an outflow of East Antarctic ice onto the West Antarctic continental shelf. Alternatively, the largely marine West Antarctic ice sheet may have appeared first, provided that post-Eocene continental drift allowed heavy snow precipitation over the West Antarctic island archipelago, but prevented warm South Pacific ocean currents from entering. Sea ice could then thicken and ground from surface accumulation and basal freezing, creating broad marine ice domes in shallow West Antarctic embayments. These marine ice domes could then merge with ice caps on the islands to create a marine ice sheet that could expand into East Antarctica and merge with highland ice caps to form the terrestrial East Antarctic ice sheet. A Cenozoic glacial history of Antarctica is outlined, based on this marine ice transgression hypothesis.