We search for single-photon decays of the Υ(1S) resonance, Υ→γ+invisible, where the invisible state is either a particle of definite mass, such as a light Higgs boson A0, or a pair of dark matter particles, χχ̄. Both A0 and χ are assumed to have zero spin. We tag Υ(1S) decays with a dipion transition Υ(2S) →π⊃+π⊃-Υ(1S) and look for events with a single energetic photon and significant missing energy. We find no evidence for such processes in the mass range mA0≤9.2GeV and mχ≤4.5GeV in the sample of 98×106 Υ(2S) decays collected with the BABAR detector and set stringent limits on new physics models that contain light dark matter states. © 2011 American Physical Society.