A proof is given for the existence and uniqueness of a stationary vacuum solution (M, g, ξ) of the boundary value problem consisting of Einstein's equations in an exterior domain M diffeomorphic to R × Σ (where Σ = R3\B(0, R)) and boundary data depending on the Killing field ξ on ∂Σ. The boundary data must be sufficiently close to that of a stationary, spatially conformally flat vacuum solutio
