  We report a search for a dark photon using $14.9$~fb$^{-1}$ of $e^+e^-$
annihilation data taken at center-of-mass energies from 4.13 to 4.60~GeV with
the BESIII detector operated at the BEPCII storage ring. The dark photon is
assumed to be produced in the radiative annihilation process of $e^+e^-$ and to
predominantly decay into light dark matter particles, which escape from the
detector undetected. The mass range from 1.5 to 2.9~GeV is scanned for the dark
photon candidate, and no significant signal is observed. The mass dependent
upper limits at the 90$\%$ confidence level on the coupling strength parameter
$\epsilon$ for a dark photon coupling with an ordinary photon vary between
$1.6\times 10^{-3}$ and $5.7\times10^{-3}$.
