Abundant geomorphic evidence for fluvial processes on the surface of Mars suggests that during the era of heavy bombardment, Mars's atmospheric pressure was high enough for liquid water to flow on the surface. Many authors have proposed mechanisms by which Mars could have lost (or sequestered) an earlier, thicker atmosphere but none of these proposals has gained general acceptance. Here we examine the process of atmospheric erosion by impacts and show that it may account for an early episode of atmosphere loss from Mars. On the basis of this model, the primordial atmospheric pressure on Mars must have been in the vicinity of 1 bar, barring other sources or sinks of CO2. Current impact fluxes are too small to erode significantly the present martian atmosphere.