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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 ϵ for a dark photon coupling with an ordinary photon vary between 1.6×10−3 and 5.7×10−3.