A modification of the statistical associating fluid theory has recently been developed to model non-conformal fluids with attractive potentials of variable range (SAFT-VR) which gives a very good description of the phase behaviour of water and its mixtures with nonelectrolytes. In the present paper we extend the SAFT-VR approach to deal with strong-electrolyte solutions (SAFT-VRE). The water molecules are modeled as hard spheres with four attractive short-range sites to describe the hydrogen-bonding association. The electrolyte molecules are modeled with two hard spheres of different size which describe the anion and cation respectively. The mean-spherical approximation (MSA) is used for the restricted primitive model (RPM) to account for the long-range Coulombic ion-ion interactions, while the long-range water-water and ion-water attractive interactions are modeled as square-well dispersive interactions treated via a second-order high-temperature expansion in the spirit of the SAFT-VR approach. We have studied nine single-salt aqueous solutions and one mixed-salt system of characteristic strong electrolytes (alkali halides) in the temperature range between 273 and 373 K. Using only one transferable fitted parameter per ion, the experimental vapor pressures and densities are very well described by the SAFT-VRE theory. As a limit of the MSA, the Debye-Huckel (DH) expression is used to describe the ion-ion interactions in one of the solutions. Due to the excellent description of the solvent in the SAFT-VR approach, the experimental vapor pressure for an aqueous solution of sodium chloride is also very well described with this simple approach.