In connection with recent Galileo images of the Jovian satellite Europa, there has been a significantly increased interest in the subglacial Lake Vostok in central East Antarctica. Since the theoretical prediction by Zotikov <1961> of the existence of lakes (one of which later was named Vostok) under ~4 km of ice in Antarctica and its confirmation by radar measurements from aircraft in 1974 <Robin et al., 1977> and from the ERS-1 satellite in 1993, it was tacitly assumed that the lake originated from the basal melting of ice Zotikov, 1961, 1977, 1986; Kapitsa et al., 1996>. Recently, Zotikov and Duxbury <2000> proposed an alternative hypothesis that if Lake Vostok had existed on the surface before the Antarctic perennial glaciation began 5--30 Myr ago, then it was buried under the thermally protecting ice and never froze all the way to the bottom. In this study we support our hypothesis by comprehensive numerical modeling and evaluate its interesting implications for Mars and exobiology. Our numerical experiments on the sensitivity of the freezing depth to different upper boundary temperatures and different internal heat flows and the recent Mars Global Surveyor Mars Orbiter Laser Altimeter (MOLA) data support our model of the origin of possible subpolar water on Mars versus the old basal melting hypothesis. We also propose here experimental tests for our hypothesis. As a result of our computations, we conclude that Lake Vostok has survived the Antarctic glaciation without freezing completely, if it were initially an open lake with a surprisingly small water depth (greater than ~53 m). The current water depth at a single point is as much as 500 m <Verkulich et al., 1996>. This maximum freezing depth of ~53 m was reached in only 3300 years when the total upper ice cover was only ~600 m. The computed critical water depth is applicable to other Antarctic lakes under a thick ice cover. For the Martian internal heat flow we have computed the critical water depth to be ~63 m. The existence of the lake before the accumulation of ice in Antarctica implies that possible microorganisms and their remnants in the lake can be older than the Antarctic ice sheet, i.e., older than 5--30 Myr. If future observations find water under the Martian polar caps, this subpolar water, according to our model, has originated from lakes that were on the Martian surface before the formation of Mars' polar caps and have not completely frozen. Such environments would be ideal candidates for extant or extinct life on the red planet. This argues for a drilling mission at the Martian polar caps. ¿ 2001 American Geophysical Union