  We propose a search for dark photons $A^{\prime}$ at the LHCb experiment
using the charm meson decay $D^*(2007)^0 \!\to D^0 A^{\prime}$. At nominal
luminosity, $D^{*0} \!\to D^0 \gamma$ decays will be produced at about 700kHz
within the LHCb acceptance, yielding over 5 trillion such decays during Run 3
of the LHC. Replacing the photon with a kinetically-mixed dark photon, LHCb is
then sensitive to dark photons that decay as $A^{\prime}\!\to e^+e^-$. We
pursue two search strategies in this paper. The displaced strategy takes
advantage of the large Lorentz boost of the dark photon and the excellent
vertex resolution of LHCb, yielding a nearly background-free search when the
$A^{\prime}$ decay vertex is significantly displaced from the proton-proton
primary vertex. The resonant strategy takes advantage of the large event rate
for $D^{*0} \!\to D^0 A^{\prime}$ and the excellent invariant mass resolution
of LHCb, yielding a background-limited search that nevertheless covers a
significant portion of the $A^{\prime}$ parameter space. Both search strategies
rely on the planned upgrade to a triggerless-readout system at LHCb in Run 3,
which will permit identification of low-momentum electron-positron pairs online
during data taking. For dark photon masses below about 100MeV, LHCb can explore
nearly all of the dark photon parameter space between existing
prompt-$A^{\prime}$ and beam-dump limits.
