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We measure the mass difference, Δm_{+}, between the D^{*}(2010)^{+} and the D^{+} using the decay chain D^{*}(2010)^{+}→D^{+}π^{0} with D^{+}→K^{-}π^{+}π^{+}. 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 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 Δm_{0}≡m(D^{*}(2010)^{+})-m(D^{0}) to obtain Δm_{D}=m(D^{+})-m(D^{0})=(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.