We report on a search for single-photon events in 53 $\rm fb^{-1}$ of data collected with the BABAR detector. We look for events consistent with production of a dark photon ($A^\prime$) through the process $e^+e^-$$\rightarrow\gamma$ $A^\prime$,$~A^\prime \rightarrow ~invisible$. Such particles are motivated by the possible existence of a $U(1) $ gauge charge for dark matter. We find no evidence for this process and set limits on the $A^\prime$-photon coupling for a dark photon mass below $8~\rm GeV/c^2$. These results greatly improve upon previous bounds, and exclude the range of values suggested by the dark-photon interpretation of the muon ($\rm g-2$) anomaly.