  We measure the mass difference, $\Delta m_+$, between the $D^{*}(2010)^+$ and
the $D^+$ using the decay chain $D^{*}(2010)^+\to D^+ \pi^0$ with $D^+\to
K^-\pi^+\pi^+$. The data were recorded with the BABAR detector at
center-of-mass energies at and near the $\Upsilon(4S)$ resonance, and
correspond to an integrated luminosity of approximately 468 ${\rm fb}^{-1}$. We
measure $\Delta m_+ = \left(140\,601.0 \pm 6.8[{\rm stat}] \pm 12.9[{\rm
syst}]\right)$ keV. We combine this result with a previous BABAR measurement of
$\Delta m_0\equiv m(D^{*}(2010)^+) - m (D^0)$ to obtain $\Delta m_D = m(D^+) -
m(D^0) = \left(4\,824.9 \pm 6.8[{\rm stat}] \pm 12.9[{\rm syst}]\right)$ keV.
These results are compatible with and approximately five times more precise
than the Particle Data Group averages.
