We report results of quark masses in quenched lattice QCD with the Kogut-Susskind fermion action, employing the regularization independent scheme of Martinelli et al. to nonperturbatively evaluate the renormalization factor relating the bare quark mass on the lattice to that in the continuum. Calculations are carried out at β = 6.0, 6.2, and 6.4, from which we find mudMS̅(2 GeV) = 4.23(29) MeV for the average up and down quark mass and, with the φ meson mass as input, msMS̅(2 GeV) = 129(12) MeV for the strange mass in the continuum limit. These values are about 20-0x1.fc49p+0rger than those obtained with the one-loop perturbative renormalization factor.