The electron-stimulated sputtering of thin amorphous solid water films deposited on Pt(111) is investigated. The sputtering appears to be dominated by two processes: (1) electron-stimulated desorption of water molecules and (2) electron-stimulated reactions leading to the production of molecular hydrogen and molecular oxygen. The electron-stimulated desorption of water increases monotonically with increasing film thickness. In contrast, the total sputtering--which includes all electron-stimulated reaction channels--is maximized for films of intermediate thickness. The sputtering yield versus thickness indicates that erosion of the film occurs due to reactions at both the water/vacuum interface and the Pt/water interface. Experiments with layered films of D2O and H2O demonstrate significant loss of hydrogen due to reactions at the Pt/water interface. The electron-stimulated sputtering is independent of temperature below approximately 80 K and increases rapidly at higher temperatures.