We measure the mass difference, Δ m 0 , between the D ∗ ( 2010 ) + and the D 0 and the natural linewidth, Γ , of the transition D ∗ ( 2010 ) + → D 0 π + . The data were recorded with the BABAR detector at center-of-mass energies at and near the Υ ( 4 S ) resonance, and correspond to an integrated luminosity of approximately 477     fb − 1 . The D 0 is reconstructed in the decay modes D 0 → K − π + and D 0 → K − π + π − π + . For the decay mode D 0 → K − π + we obtain Γ = ( 83.4 ± 1.7 ± 1.5 )     keV and Δ m 0 = ( 145425.6 ± 0.6 ± 1.8 )     keV , where the quoted errors are statistical and systematic, respectively. For the D 0 → K − π + π − π + mode we obtain Γ = ( 83.2 ± 1.5 ± 2.6 )     keV and Δ m 0 = ( 145426.6 ± 0.5 ± 2.0 )     keV . The combined measurements yield Γ = ( 83.3 ± 1.2 ± 1.4 )     keV and Δ m 0 = ( 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.