An Exact Black Hole Scattering Amplitude

General Relativity famously predicts precession of orbital motions in the Schwarzschild metric. In this paper we show that by adding a NUT charge $N=iM$ the precession vanishes to all orders in $G$ even for rotating black holes. Moreover, we conjecture a generalization of the eikonal formula and show that the classical integrable trajectories determine the full quantum amplitude for this black hole, by means of exponentiation of the Post-Minkowskian radial action. Several consequences of integrability in self-dual gravity are discussed.