Abstract: We search for single-photon decays of the Gamma(1S) resonance, Gamma -> gamma + invisible, where the invisible state is either a particle of definite mass, such as a light Higgs boson A(0), or a pair of dark matter particles, chi(chi) over bar. Both A(0) and chi are assumed to have zero spin. We tag Gamma(1S) decays with a dipion transition Gamma(1S) -> pi(+)pi(-)Y(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 m(A0) <= 9.2 GeV and m(chi) <= 4.5 GeV in the sample of 98 x 10(6) Gamma(2S) decays collected with the BABAR detector and set stringent limits on new physics models that contain light dark matter states.