Neutron spectroscopy data acquired by Mars Odyssey are analyzed to determine the abundance and depth of near-surface water ice as a function of latitude in the southern hemisphere as well as the inventory of CO2 in the south polar residual cap. The surface is modeled as a semi-infinite, water-rich permafrost layer covered by desiccated material, which is consistent with theoretical models of ground ice stability. Latitude-dependent parameters, water abundance and depth, are determined from zonally averaged neutron counting data. Spatial mixing of the output of neutrons from regions within the footprint of the spectrometer is modeled, and asymmetrical features such as the residual cap are included in the analysis. Absorption of thermal neutrons by major elements other than hydrogen is found to have a significant influence on the determination of water abundance. Poleward of -60¿, the water-rich layer contains 60% ¿ 10% water by weight (70% to 85% by volume) and is covered by less than 15 g/cm2 ¿ 5 g/cm2 of dry material. The volume fraction of water is generally higher than can be accommodated in the pore space of surface soils, which implies that water vapor diffusion processes alone cannot explain the observations. Alternatives for the formation of the water-rich layer are discussed. Results of our analysis of the residual-cap CO2 inventory support conclusions that the atmosphere is not buffered by a larger reservoir of surface CO2 at the poles and that Mars' total CO2 inventory is well represented by the present atmospheric mass.