We report a measurement of the D0 meson mass using the decay chain D*(2010)+ -> D0 pi+ with D0 -> K- K- K+ pi+. The data were recorded with the BaBar detector at center-of-mass energies at and near the Y(4S) resonance, and correspond to an integrated luminosity of approximately 477 1/fb. We obtain m(D0) = (1864.841 +- 0.048 +- 0.063) MeV, where the quoted errors are statistical and systematic, respectively. The uncertainty of this measurement is half that of the best previous measurement.