The possibility that the Earth suffered episodes of global glaciation as recently as the Neoproterozoic period, between about 900 and 543 million years ago, has been widely discussed(1-3). Termination of such 'hard snowball Earth' climate states has been proposed to proceed from accumulation of carbon dioxide in the atmosphere(4). Many salient aspects of the snowball scenario depend critically on the threshold of atmospheric carbon dioxide concentrations needed to trigger deglaciation(2,5). Here I present simulations with a general circulation model, using elevated carbon dioxide levels to estimate this deglaciation threshold. The model simulates several phenomena that are expected to be significant in a 'snowball Earth' scenario, but which have not been considered in previous studies with less sophisticated models, such as a reduction of vertical temperature gradients in winter, a reduction in summer tropopause height, the effect of snow cover and a reduction in cloud greenhouse effects. In my simulations, the system remains far short of deglaciation even at atmospheric carbon dioxide concentrations of 550 times the present levels (0.2 bar of CO2). I find that at much higher carbon dioxide levels, deglaciation is unlikely unless unknown feedback cycles that are not captured in the model come into effect.