Various lines of geomorphic evidence suggest that large bodies of groundwater were present on Mars throughout much of its early geologic history. In this paper the hydrologic response of groundwater to the thermal evolution of the early martian crust is considered. When a temperature gradient is present in a moist porous medium, it gives rise to a vapor pressure gradient that drives the diffusion of water vapor from regions of high to low temperature. By this process, a geothermal gradient as small as 15 K km-1 could drive the vertical transport of 1 km of water to the freezing front at the base of the martian cryosphere every 106 -- 107 years, or the equivalent of ~102 -- 103 km of water over the course of martian geologic history. Models of the thermal history of Mars suggest that this thermally-driven vapor flux may have been as much as 3--5 times greater in the past. The magnitude of this transport suggests that the process of geothermally-induced vapor diffusion may have played a critical role in the initial emplacement of ground ice and the subsequent geomorphic and geochemical evolution of the martian crust. ¿American Geophysical Union 1991