We present a model for the growth of frazil ice crystals and their accumulation as marine ice at the base of Antarctic ice shelves. The model describes the flow of buoyant water upward along the ice shelf base and includes the differential growth of a range of crystal sizes. Frazil ice formation starts when the rising plume becomes supercooled. Initially, the majority of crystals have a radius of similar to0.3 mm and concentrations are below 0.1 g/L. Depending on the ice shelf slope, which controls the plume speed, frazil crystals increase in size and number. Typically, crystals up to 1.0 mm in radius are kept in suspension, and concentrations reach a maximum of 0.4 g/L. The frazil ice in suspension decreases the plume density and thus increases the plume speed. Larger crystals precipitate upward onto the ice shelf base first, with smaller crystals following as the plume slows down. In this way, marine ice is formed at rates of up to 4 m/yr in some places, consistent with areas of observed basal accumulation on Filchner-Ronne Ice Shelf. The plume continues below the ice shelf as long as it is buoyant. If the plume reaches the ice front, its rapid rise produces high supercooling and the ice crystals attain a radius of several millimeters before reaching the surface. Similar ice crystals have been trawled at depth north of Antarctic ice shelves, but otherwise no observations exist to verify these first predictions of ice crystal sizes and volumes.