  We consider the high energy limit of the colour ordered one-loop five-gluon
amplitude in the planar maximally supersymmetric N=4 Yang-Mills theory in the
multi-Regge kinematics where all of the gluons are strongly ordered in
rapidity. We apply the calculation of the one-loop pentagon in D=6-2 eps
performed in a companion paper to compute the one-loop five-gluon amplitude
through to O(eps^2). Using the factorisation properties of the amplitude in the
high-energy limit, we extract the one-loop gluon-production vertex to the same
accuracy, and, by exploiting the iterative structure of the gluon-production
vertex implied by the BDS ansatz, we perform the first computation of the
two-loop gluon-production vertex up to and including finite terms.
