Numerical calculations of the equation of state are made for the one-dimensional gas model proposed and discussed by Ikeda in the preceding papers I and II. The model consists of molecules with hard cores, short-range (square-potential) repulsions and infinitely-long-range attractions (given by the potential -2 a / L , a =positive constant, L =length of the gas). For this model it is exactly shown that, for sufficiently weak attractions, the gas-liquid and liquid-solid transitions occur at all sufficiently low temperatures (as in helium), and that, in a certain range of strengths of attraction, only the gas-solid transition occurs at all temperatures lower than T t (=triple point) and the gas-liquid and liquid-solid transitions occur at temperatures higher than T t but lower than a certain temperature (as in usual substances in nature), and that, for sufficiently strong attractions, only one transition occurs, as in a van der Waals gas.