The problem of deglaciating a globally ice-covered (hard snowball) Earth is examined using a series of general circulation model simulations. The aim is to determine the amount of CO2 that must be accumulated in the atmosphere in order to trigger deglaciation. Prior treatments of this problem have been limited to energy balance models, which are incapable of treating certain crucial physical processes that turn out to strongly affect the conditions under which deglaciation can occur. CO2 concentrations up to .2 bars are considered in the general circulation model simulations, and even at such high CO2 content the model radiation code is found to perform well in comparison with codes explicitly designed for high CO2. In contrast to prevailing expectations, the hard snowball Earth is found to be nearly 30 K short of deglaciation, even at .2 bars. The very cold climates arise from a combination of the extreme seasonal and diurnal cycle, lapse rate effects, snow cover, and weak cloud effects. Several aspects of the atmospheric dynamics are examined in detail. The simulations indicate that the standard scenario, wherein snowball termination occurs after a few tenths of a bar of CO2 has built up following cessation of weathering, is problematic. However, the climate was found to be sensitive to details of a number of parameterized physical processes, notably clouds and heat transfer through the stable boundary layer. It is not out of the question that other parameterization suites might permit deglaciation. The results should not be construed as meaning that the hard snowball state could not have occurred, but only that deglaciation requires the operation of as-yet undiscovered processes that would enhance the climate sensitivity. A brief survey of some of the possibilities is provided.