We present the two-loop pure gauge contribution to the gluon-gluon scatteri
ng amplitude with maximal helicity violation. Our construction of the ampli
tude does not rely directly on Feynman diagrams, but instead uses its analy
tic properties in 4 - 2 epsilon dimensions. We evaluate the loop integrals
appearing in the amplitude through O(epsilon (0)) in terms of polylogarithm
s.