We present results for the light quark masses for the Wilson quark action obtained with the PCAC relation for the one-link extended axial vector current in quenched QCD at $beta=5.9-6.5$. This method leads to a remarkable improvement of scaling behavior of the light quark masses compared to the conventional method. We obtain ${bar m}_l=3.87(37)$MeV for the averaged up and down quark mass and ${bar m}_s=97(9)$MeV for the strange quark mass in the ${barMS}$ scheme at $mu=2$GeV.