Ice sheets may have reached the Equator in the late Proterozoic era (600-800Myr ago), according to geological and palaeomagnetic studies, possibly resulting in a `snowball Earth'. But this period was a critical time in the evolution of multicellular animals, posing the question of how early life survived under such environmental stress. Here we present computer simulations of this unusual climate stage with a coupled climate/ice-sheet model. To simulate a snowball Earth, we use only a reduction in the solar constant compared to present-day conditions and we keep atmospheric CO2 concentrations near present levels. We find rapid transitions into and out of full glaciation that are consistent with the geological evidence. When we combine these results with a general circulation model, some of the simulations result in an equatorial belt of open water that may have provided a refugium for multicellular animals.