We search for a low-mass scalar CP-odd Higgs boson, A0, produced in the radiative decay of the upsilon resonance and decaying into a τ+τ- pair: Υ(1S)→γA0. The production of Υ(1S) mesons is tagged by Υ(2S) →π+π-Υ(1S) transitions, using a sample of (98.3±0.9)×106 Υ(2S) mesons collected by the BABAR detector. We find no evidence for a Higgs boson in the mass range 3.5≤m A0≤9.2 GeV, and combine these results with our previous search for the tau decays of the light Higgs in radiative Υ(3S) decays, setting limits on the coupling of A0 to the bb̄ quarks in the range 0.09-1.9. Our measurements improve the constraints on the parameters of the next-to-minimal-supersymmetric Standard Model and similar theories with low-mass scalar degrees of freedom. © 2013 American Physical Society.