A theoretical method for calculating the thermodynamic properties and phase equilibria of a binary liquid mixture using the reference interaction-site model (RISM) integral equation theory, which we had proposed recently, was extended to ternary liquid systems containing salt. A novel dielectric correction of the RISM theory for a mixture of solvents was also proposed. The theory was applied to mixtures composed of water, alcohol, and NaCl, where the alcohol was either methanol or ethanol. The decrease in NaCl solubility with increasing alcohol molar fractions in the solvent was calculated. In the ethanol system, the theory yielded salt-induced liquid-liquid phase separation, which was observed experimentally in a ternary mixture of water, 1-propanol, and NaCl. The phase diagram of the ternary system was determined theoretically.