  We describe a systematic framework for computing the conservative potential
of a compact binary system using modern tools from scattering amplitudes and
effective field theory. Our approach combines methods for integration and
matching adapted from effective field theory, generalized unitarity, and the
double-copy construction, which relates gravity integrands to simpler
gauge-theory expressions. With these methods we derive the third
post-Minkowskian correction to the conservative two-body Hamiltonian for
spinless black holes. We describe in some detail various checks of our
integration methods and the resulting Hamiltonian.
