A search is presented for the decay Υ (1S) → γA 0 , A 0 → cc, where A 0 is a candidate for the CP -odd Higgs boson of the next-to-minimal supersymmetric standard model. The search is based on data collected with the BABAR detector at the Υ (2S) resonance. A sample of Υ (1S) mesons is selected via the decay Υ (2S) → π + π − Υ (1S). The A 0 → cc decay is identified through the reconstruction of hadronic D 0 , D + , and D * (2010) + meson decays. No significant signal is observed. The measured 90% confidence-level upper limits on the product branching fraction B(Υ (1S) → γA 0 ) × B(A 0 → cc) range from 7.4 × 10 −5 to 2.4 × 10 −3 for A 0 masses from 4.00 to 8.95 GeV/c 2 and 9.10 to 9.25 GeV/c 2 , where the region between 8.95 and 9.10 GeV/c 2 is excluded because of background from Υ (2S) → γχ bJ (1P ), χ bJ (1P ) → γΥ (1S) decays.