The ice-rich south polar layered deposits of Mars were probed with the Mars Advanced Radar for Subsurface and Ionospheric Sounding on the Mars Express orbiter. The radar signals penetrate deep into the deposits (more than 3.7 kilometers). For most of the area, a reflection is detected at a time delay that is consistent with an interface between the deposits and the substrate. The reflected power from this interface indicates minimal attenuation of the signal, suggesting a composition of nearly pure water ice. Maps were generated of the topography of the basal interface and the thickness of the layered deposits. A set of buried depressions is seen within 300 kilometers of the pole. The thickness map shows an asymmetric distribution of the deposits and regions of anomalous thickness. The total volume is estimated to be 1.6 × 106 cubic kilometers, which is equivalent to a global water layer approximately 11 meters thick.