We measure the mass difference, Δm+, between the D∗(2010)+ and the D+ using the decay chain D∗(2010)+→D+π0 with D+→K-π+π+. The data were recorded with the BABAR detector at center-of-mass energies at and near the I(4S) resonance, and correspond to an integrated luminosity of approximately 468 fb-1. We measure Δm+=(140 601.0±6.8[stat]±12.9[syst]) keV. We combine this result with a previous BABAR measurement of Δm0≡m(D∗(2010)+)-m(D0) to obtain ΔmD=m(D+)-m(D0)=(4824.9±6.8[stat]±12.9[syst]) keV. These results are compatible with and approximately five times more precise than the Particle Data Group averages.