Two-loop amplitudes for gluon fusion into two photons

We present the two-loop matrix elements for the scattering of two gluons into two photons in QCD. These matrix elements will enter into improved estimates of the QCD background to Higgs production at the LHC when the Higgs decays into two photons. The photon mode is especially important if M{sub H} < 140 GeV. We also give the amplitudes for the crossed process, g{gamma} {yields} g{gamma}.