An ice pump is a heat engine, driven by the change of freezing point with pressure, which will melt ice at depth in the ocean and deposit it at a shallower location: it is self-starting. Calculations of the maximum magnitude of this effect are made which show good agreement with field data available for sea and lake ice. The discussion is applied to the general case of a moving pack ice sheet with a well-mixed surface layer and to floating ice shelves. The rate of melt from an 11-m-deep pressure ridge keel due to ice pumping is estimated as 26 cm/year, and that from the front of the Ross Ice Shelf at McMurdo Sound, Antarctica is estimated as 5 m/year for the level of water movement noted in the authors' field observations. Far from the ice front, pumping between shelf areas of different thickness will still occur, with tidal motion providing the necessary water exchange, but its magnitude is now limited by the ability to remove the potentially stable layer of melt water out of the system. It is important to realize that the pumping does not depend on the availability of sensible heat in the water column and its effects are additional to any melting caused by the advection of warmer water to the ice-water interface.