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 m_{A'} ≤ 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.