The 36Ar-40Ar vapor pressure isotope effect in the liquid and solid phases has been experimentally determined at about 100 temperatures between 83 and 97 K using a high-accuracy double-differential manometric technique. Differences between the vapor pressure of a highly enriched sample of 36Ar and that of natural argon were measured simultaneously with the absolute vapor pressure of natural argon. Absolute and differential vapor pressure measurements show an estimated accuracy of 0.15% and 0.001%, respectively, with respect to the absolute vapor pressure of argon. Data were obtained in both the solid and liquid regions. The vapor pressure of 36Ar is always higher than that of 40Ar by about 0.5%. Triple-point temperatures and pressures were also measured for both isotopes. All data compare favorably with previous results reported in the literature and are nicely interpreted within the framework of Bigeleisen's theory of isotope effects. Using a large temperature extrapolation of our data, liquid-vapor isotope fractionation factors were successfully inferred from the vapor pressure measurements; the estimated values agree with the experimental ones up to close to the critical point. The isotopic difference in the molar enthalpies of vaporization and sublimation was also calculated from the vapor pressure data. Additionally, molar volume and second virial coefficient isotope effects are estimated using several different theoretical approaches.