  We calculate the set of O(\alpha_s) corrections to the double differential
decay width d\Gamma_{77}/(ds_1 \, ds_2) for the process \bar{B} \to X_s \gamma
\gamma originating from diagrams involving the electromagnetic dipole operator
O_7. The kinematical variables s_1 and s_2 are defined as s_i=(p_b -
q_i)^2/m_b^2, where p_b, q_1, q_2 are the momenta of b-quark and two photons.
While the (renormalized) virtual corrections are worked out exactly for a
certain range of s_1 and s_2, we retain in the gluon bremsstrahlung process
only the leading power w.r.t. the (normalized) hadronic mass
s_3=(p_b-q_1-q_2)^2/m_b^2 in the underlying triple differential decay width
d\Gamma_{77}/(ds_1 ds_2 ds_3). The double differential decay width, based on
this approximation, is free of infrared- and collinear singularities when
combining virtual- and bremsstrahlung corrections. The corresponding results
are obtained analytically. When retaining all powers in s_3, the sum of
virtual- and bremstrahlung corrections contains uncanceled 1/\epsilon
singularities (which are due to collinear photon emission from the s-quark) and
other concepts, which go beyond perturbation theory, like parton fragmentation
functions of a quark or a gluon into a photon, are needed which is beyond the
scope of our paper.
