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We study thermodynamic properties of an axial next-nearest-neighbor Ising model in two dimensions using a cluster heat bath Monte Carlo method. The method drastically reduces the relaxation time, and enables us to study equilibrium properties of the model. We show that the model really exhibits two phase transitions at T c1 and T c2 (ϽT c1 ), as predicted by a free fermion approximation. The one at T c1 is shown to be of the Kosterlitz-Thouless type, and the other to be of the Pokrovsky-Talapov type. The higher transition temperature T c1 is considerably lower than those estimated by previous authors. Low-temperature properties as well as the lower transition temperature T c2 are found to be well described by the fermion approximation.