The enthalpy, the heat capacity, and Gibbs free energy of glassy water and of metastable water at 153 K have been examined by measuring the heat evolved on its crystallization to cubic ice. Measurements are made both isothermally and for slow heating, since the total heat evolved on crystallization decreases with the temperature. It is shown that a small fraction of metastable water persists up to 180 K when heated at 30 K min(-1) and that the excess enthalpy of water at 153 K is 1.2 +/- 0.1 kJ mol(-1). Free energy considerations suggest that a thermodynamic continuity is possible only if the residual entropy of glassy water at 0 K is less than or equal to 5.3 J K-1 mol(-1), or the excess residual entropy over that of hexagonal ice is less than or equal to 1.9 J K-1 mol(-1). Precise estimates have been made by finding a variation of heat capacity with temperature between 153 and 236 K which satisfy the requirements of both the enthalpy and the third law of thermodynamics. These give a value of 3.8-4.5 +/- 0.1 J K-1 mol(-1) for the residual entropy of glassy water, or 0.4-1.1 +/- 0.1 J K-1 mol(-1) more than that of hexagonal ice. On this basis, it is concluded that glassy water and normal water can be connected by a continuous thermodynamic path. Recent estimates of the excess residual entropy of water are consistent with this value. Its absolute value can be determined only when its reversible transformation to a state of known entropy becomes observable.