A theoretical explanation of the experimentally observed characteristic thermal anomalies in the specific heat of lipid bilayers containing cholesterol is provided in terms of the phase equilibria in the phosphatidylcholine-cholesterol system. The phase equilibria are calculated via a microscopic interaction model that takes proper account of both the conformational and the crystalline degrees of freedom of the lipid acyl chains. It is shown that the characteristic double-peaked specific heat, with a narrow and a broad component, is a natural consequence of the topology of the phase diagram. Some results for the enthalpy of the mixed system are also reported. It is suggested that there is no need for invoking special mechanisms such as lipid-cholesterol complexing or formation of special interfacial regions in the bilayer in order to explain the specific-heat anomalies.