  We present the amplitude for classical scattering of gravitationally
interacting massive scalars at third post-Minkowskian order. Our approach
harnesses powerful tools from the modern amplitudes program such as generalized
unitarity and the double-copy construction, which relates gravity integrands to
simpler gauge-theory expressions. Adapting methods for integration and matching
from effective field theory, we extract the conservative Hamiltonian for
compact spinless binaries at third post-Minkowskian order. The resulting
Hamiltonian is in complete agreement with corresponding terms in
state-of-the-art expressions at fourth post-Newtonian order as well as the
probe limit at all orders in velocity. We also derive the scattering angle at
third post-Minkowskian order.
