We search for single-photon events in 53 fb-1 of e+e- collision data collected with the BABAR detector at the PEP-II B-Factory. We look for events with a single high-energy photon and a large missing momentum and energy, consistent with production of a spin-1 particle A′ through the process e+e-→γA′; A′→invisible. Such particles, referred to as "dark photons," are motivated by theories applying a U(1) gauge symmetry to dark matter. We find no evidence for such processes and set 90% confidence level upper limits on the coupling strength of A′ to e+e- in the mass range mA′≤8 GeV. In particular, our limits exclude the values of the A′ coupling suggested by the dark-photon interpretation of the muon (g-2)μ anomaly, as well as a broad range of parameters for the dark-sector models.