We confirm by explicit computation the conjectured all-orders iteration of planar maximally supersymmetric ${\cal N}=4$ Yang-Mills theory in the nontrivial case of five-point two-loop amplitudes. We compute the required unitarity cuts of the integrand and evaluate the resulting integrals numerically using a Mellin--Barnes representation and the automated package of M.~Czakon (hep-ph/0511200). This confirmation of the iteration relation provides further evidence suggesting that ${\cal N}=4$ gauge theory is solvable.