  We consider a typical hard scattering process in a heat bath of photons and
electrons at temperature, $T$, in finite temperature QED. We show that, when
the hard scattering scale is much larger than the temperature, the infrared
pieces of both the real and virtual parts of the cross section factorise; these
can be exponentiated and cancel between each other to all orders in
perturbation theory. We use the technique of Grammer and Yennie to give a
prescription for the extraction of the infrared divergent parts, and for the
form of the finite remainder. Symmetry arguments are used to show the
finiteness of new terms arising in the $T \neq 0$ part of the computation.
