We measure the mass difference Δm0 between the D*(2010)+ and the D0 and the natural linewidth Γ of the transition D*(2010)+ → D0π+. The data were recorded with the BABAR detector at center-of-mass energies at and near the Υ(4S) resonance, and correspond to an integrated luminosity of approximately 477 fb(-1). The D0 is reconstructed in the decay modes D0 → K- π+ and D0 → K- π+ π- π+. For the decay mode D0 → K- π+ we obtain Γ = (83.4±1.7±1.5) keV and Δm0 = (145425.6±0.6±1.7) keV, [corrected] where the quoted errors are statistical and systematic, respectively. For the D0 → K- π+ π- π+ mode we obtain Γ = (83.2±1.5±2.6) keV and Δm0 = (145426.6±0.5±1.9) keV. [corrected] The combined measurements yield Γ = (83.3±1.2±1.4) keV and Δm0 = (145425.9±0.4±1.7) keV; the width is a factor of approximately 12 times more precise than the previous value, while the mass difference is a factor of approximately 6 times more precise.