In this study, we use a cloud model to simulate dehydration which occurs due to formation of optically thin, laminar cirrus as air rises slowly across the tropopause. The slow ascent and adiabatic cooling, which balances the radiative heating near the tropopause, drives nucleation of a very small number of ice crystals (<1L-1). These crystals grow rapidly and sediment out within a few hours. The clouds never become optically thick enough to be visible from the ground. The ice crystal nucleation and growth prevents the relative humidity with respect to ice (RHI) from rising more than a few percent above the threshold for ice nucleation (RHInuc≃110--160%, depending upon the aerosol composition); hence, laminar cirrus can limit the mixing ratio of water vapor entering the stratosphere. However, the ice number densities are too low and their sedimentation is too rapid to allow dehydration of the air from RHInuc down to saturation (RHI=100%). The net result is that air crosses the tropopause with water vapor mixing ratios about 1.1 to 1.6 times the ice saturation mixing ratio corresponding to the tropopause temperature, depending on the threshold of ice nucleation on aerosols in the tropopause region. If the cross-tropopause ascent rate is larger than that calculated to balance radiative heating (0.2 cm s-1), then larger ice crystal number densities are generated, and more effective dehydration is possible (assuming a fixed temperature). The water vapor mixing ratio entering the stratosphere decreases with increasing ascent rate (approaching the tropopause ice saturation mixing ratio) until the vertical wind speed exceeds the ice crystal terminal velocity (about 10 cm s-1). More effective dehydration can also be provided by temperature oscillations associated with wave motions. The water vapor mixing ratio entering the stratosphere is essentially controlled by the tropopause temperature at the coldest point in the wave. Hence, the efficiency of dehydration at the tropopause depends upon both the effectiveness of upper tropospheric aerosols as ice nuclei and the occurrence of wave motions in the tropopause region. In situ humidity observations from tropical aircraft campaigns and balloon launches over the past several years have provided a few examples of ice-supersaturated air near the tropopause. However, given the scarcity of data and the uncertainties in water vapor measurements, we lack definitive evidence that air entering the stratosphere is supersaturated with respect to ice. ¿ 2001 American Geophysical Union