We present a search for a Higgs boson decaying into a pair of photons based on $9.6\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$ collected with the D0 detector at the Fermilab Tevatron Collider. The search employs multivariate techniques to discriminate signal from the nonresonant background and is separately optimized for a standard model and a fermiophobic Higgs boson. No significant excess of data above the background prediction is observed and upper limits on the product of the cross section and branching fraction are derived at the 95% confidence level as a function of Higgs boson mass. For a standard model Higgs boson with mass of 125 GeV, the observed (expected) upper limits are a factor of 12.8 (8.7) above the standard model prediction. The existence of a fermiophobic Higgs boson with mass in the 100--113 GeV range is excluded at the 95% confidence level.