The late Neoproterozoic era, ~600--800 Myr ago, was marked by at least two intervals of widespread cold that left glacial deposits at low paleolatitudes. Both Snowball solutions with global ice cover and Slushball solutions with ice-free tropical oceans have been proposed to explain the paleomagnetic data. The Snowball model is best able to explain the auxiliary geological evidence, particularly the existence of cap carbonates, but it implies drastic survival pressure on the photosynthetic biota if the ice cover was everywhere thick enough to prevent sunlight from reaching the underlying ocean. A thin ice solution that avoids this problem has been proposed but is thought to be inconsistent with realistic optical properties of sea ice and with the equatorward flow of sea glaciers. Here we use a coupled energy-balance climate/sea-glacier model to argue that these apparent difficulties can be overcome and that thin tropical ice may have prevailed during these remarkable glacial episodes. We also suggest that (1) some processes beyond the scope of zonal mean energy-balance models may significantly affect the solutions and (2) thin ice could have prevailed on low-latitude seas (or large lakes) protected from large-scale sea-glacier flow by surrounding land even if most of the Earth was in a hard Snowball state.