The configurational entropy of ice is calculated by thermodynamic integration from high to low temperatures. We use Monte Carlo simulations with a simple energy model which reproduces the Bernal Fowler ice rules. This procedure is found to be precise enough to give reliable values for the residual entropy s(th) of different ice phases in the thermodynamic limit. First, we check it for a two-dimensional ice model. Second, we calculate s(th) for ice Ih, and compare our result with those previously given in the literature. Third, we obtain s(th) for ice VI, for which we find a value clearly higher than for ice Ih. (C) 2013 Elsevier B.V. All rights reserved.