  We compute the full-color two-loop five-gluon amplitude for the all-plus
helicity configuration. In order to achieve this, we calculate the required
master integrals for all permutations of the external legs, in the physical
scattering region. We verify the expected divergence structure of the
amplitude, and extract the finite hard function. We further validate our result
by checking the factorization properties in the collinear limit. Our result is
fully analytic and valid in the physical scattering region. We express it in a
compact form containing logarithms, dilogarithms and rational functions.
