According to the Snowball Earth hypothesis, Earth has experienced periods of low-latitude glaciation in its deep past. Prior studies have used general circulation models (GCMs) to examine the effects such an extreme climate state might have on the structure and dynamics of Earth's troposphere, but the behavior of the stratosphere has not been studied in detail... Understanding the snowball stratosphere is important for developing an accurate account of the Earth's radiative and chemical properties during these episodes. Here we conduct the first analysis of the stratospheric circulation of the Snowball Earth using ECHAM6 general circulation model simulations. In order to understand the factors contributing to the stratospheric circulation, we extend the Statistical Transformed Eulerian Mean framework. We find that the stratosphere during a snowball with prescribed modern ozone levels exhibits a weaker meridional overturning circulation, reduced wave activity, stronger zonal jets, and is extremely cold relative to modern conditions. Notably, the snowball stratosphere displays no sudden stratospheric warmings. Without ozone, the stratosphere displays slightly weaker circulation, a complete lack of polar vortex, and even colder temperatures. We also explicitly quantify for the first time the cross-tropopause mass exchange rate and stratospheric mixing efficiency during the snowball and show that our values do not change the constraints on CO$_2$ inferred from geochemical proxies during the Marinoan glaciation ($\sim$635 Ma), unless the O$_2$ concentration during the snowball was orders of magnitude less than the CO$_2$ concentration. (read more)