Absence of three-loop four-point ultraviolet divergences in N=4 supergravity

We compute the coefficient of the potential three-loop four-point ultraviolet divergence in pure N=4 supergravity and show that it vanishes, contrary to expectations from symmetry arguments. The recently uncovered duality between color and kinematics is used to greatly streamline the calculation. We comment on all-loop cancellations hinting at further surprises awaiting discovery at higher loops.