  We present the O(alpha_s^4) virtual QCD corrections to gluon-gluon scattering
due to the interference of tree and two-loop amplitudes. We work in
conventional dimensional regularisation and give analytic expressions
renormalised in the MSbar scheme. The structure of the infrared divergences
agrees with that predicted by Catani while formulae for the finite remainder
are given in terms of logarithms and polylogarithms that are real in the
physical region. These results, together with those previously obtained for
quark-quark and quark-gluon scattering, complete the two-loop matrix elements
needed for the next-to-next-to-leading order contribution to inclusive jet
production at hadron colliders.
