South American glaciers, including those in Patagonia, presently contribute the largest amount of meltwater to sea level rise per unit glacier area in the world. Yet understanding of the mechanisms behind the associated glacier mass balance changes remains unquantified partly because models are hindered by a lack of knowledge of subglacial topography. This study applied a perfect-plasticity model along glacier centre-lines to derive a first-order estimate of ice thickness and then interpolated these thickness estimates across glacier areas. This produced the first complete coverage of distributed ice thickness, bed topography and volume for 617 glaciers between 41oS and 55oS and in 24 major glacier regions. Maximum modelled ice thicknesses reach 1631 m ± 179 m in the South Patagonian Icefield (SPI), 1315 m ± 145 m in the North Patagonian Icefield (NPI) and 936 m ± 103 m in Cordillera Darwin. The total modelled volume of ice is 1234.6 km3 ± 246.8 km3 for the NPI, 4326.6 km3 ± 865.2 km3 for the SPI and 151.9 km3 ± 30.38 km3 for Cordillera Darwin. The total volume was modelled to be 5955 km3 ± 1191 km3, which equates to 5458.3 Gt ± 1091.6 Gt ice and to 15.08 mm ± 3.01 mm sea level equivalent (SLE). However, a total area of 655 km2 contains ice below sea level and there are 282 individual overdeepenings with a mean depth of 38 m and a total volume if filled with water to the brim of 102 km3. Adjusting the potential SLE for the ice volume below sea level and for the maximum potential storage of meltwater in these overdeepenings produces a maximum potential sea level rise (SLR) of 14.71 mm ± 2.94 mm. We provide a calculation of the present ice volume per major river catchment and we discuss likely changes to southern South America glaciers in the future. The ice thickness and subglacial topography modelled by this study will facilitate future studies of ice dynamics and glacier isostatic adjustment, and will be important for projecting water resources and glacier hazards.