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A technique for translating the classical scattering function of twogravitationally interacting bodies into a corresponding (effective one-body)Hamiltonian description has been recently introduced [Phys.\ Rev.\ D {\bf 94},104015 (2016)]. Using this technique, we derive, for the first time, tosecond-order in Newton's constant (i.e. one classical loop) the Hamiltonian oftwo point masses having an arbitrary (possibly relativistic) relative velocity.The resulting (second post-Minkowskian) Hamiltonian is found to have a tamehigh-energy structure which we relate both to gravitational self-force studiesof large mass-ratio binary systems, and to the ultra high-energy quantumscattering results of Amati, Ciafaloni and Veneziano. We derive severalconsequences of our second post-Minkowskian Hamiltonian: (i) the need to usespecial phase-space gauges to get a tame high-energy limit; and (ii)predictions about a (rest-mass independent) linear Regge trajectory behavior ofhigh-angular-momenta, high-energy circular orbits. Ways of testing thesepredictions by dedicated numerical simulations are indicated. We finallyindicate a way to connect our classical results to the quantum gravitationalscattering amplitude of two particles, and we urge amplitude experts to usetheir novel techniques to compute the 2-loop scattering amplitude of scalarmasses, from which one could deduce the third post-Minkowskian effectiveone-body Hamiltonian.

High-energy gravitational scattering and the general relativistic two-body problem