  At hadron colliders, the leading production mechanism for a pair of photons
is from quark-anti-quark annihilation at the tree level. However, due to large
gluon-gluon luminosity, the loop-induced process $gg\to \gamma \gamma$ provides
a substantial contribution. In particular, the amplitudes mediated by the top
quark become important at the $t \bar t$ threshold and above. In this letter we
present the first complete computation of the next-to-leading order (NLO)
corrections (up to $\alpha_S^3$) to this process, including contributions from
the top quark. These entail two-loop diagrams with massive propagators whose
analytic expressions are unknown and have been evaluated numerically. We find
that the NLO corrections to the top-quark induced terms are very large at low
diphoton invariant mass $m(\gamma \gamma)$ and close to the $t \bar t$
threshold. The full result including five massless quarks and top quark
contributions at NLO displays a much more pronounced change of slope in the
$m(\gamma \gamma)$ distribution at $t \bar t$ threshold than at LO and an
enhancement at high invariant mass with respect to the massless calculation.
