Radar observations of Mercury have provided the startling discovery of the probable existence of substantial deposits of ices in permanently shaded polar regions <Butler et al., 1993; Harmon et al., 1994>. This has renewed the old argument about the existence of such deposits in the polar regions of the Moon. It is most likely necessary for volatiles to be able to migrate to the polar regions of these bodies in order to build up such deposits. This paper presents the results of the extension of a Monte Carlo simulation of the migration of molecules on the surface of Mercury by Butler et al. <1993>. The results of the simulations show that for typical conditions on Mercury, ~5--15% of all H2O which is placed randomly on the surface migrates to stable (permanently shaded) polar regions. For the Moon the numbers are ~20--50%. The numbers for the migration of CO2 are ~2% for Mercury and ~13% for the Moon. These percentages are similar to those previously calculated and support the idea of ice deposits in the polar regions of both Mercury and the Moon.¿ 1997 American Geophysical Union