Dark sectors charged under a new Abelian interaction have recently received much attention in the context of dark matter models. These models introduce a light new mediator, the so-called dark photon (A^{'}), connecting the dark sector to the standard model. We present a search for a dark photon in the reaction e^{+}e^{-}→γA^{'}, A^{'}→e^{+}e^{-}, μ^{+}μ^{-} using 514 fb^{-1} of data collected with the BABAR detector. We observe no statistically significant deviations from the standard model predictions, and we set 90% confidence level upper limits on the mixing strength between the photon and dark photon at the level of 10^{-4}-10^{-3} for dark photon masses in the range 0.02-10.2 GeV. We further constrain the range of the parameter space favored by interpretations of the discrepancy between the calculated and measured anomalous magnetic moment of the muon.