  Using scattering amplitudes, we obtain the potential contributions to
conservative binary dynamics in general relativity at fourth post-Minkowskian
order, ${\cal O}(G^4)$. As in previous lower-order calculations, we harness
powerful tools from the modern scattering amplitudes program including
generalized unitarity, the double copy, and advanced multiloop integration
methods, in combination with effective field theory. The classical amplitude
involves polylogarithms with up to transcendental weight two and elliptic
integrals. We derive the radial action directly from the amplitude, and
determine the corresponding Hamiltonian in isotropic gauge. Our results are in
agreement with known overlapping terms up to sixth post-Newtonian order, and
with the probe limit. We also determine the post-Minkowskian energy loss from
radiation emission at ${\cal O}(G^3)$ via its relation to the tail effect.
