  We show that certain solutions to the linearized Einstein equation can---by
the application of a particular type of linearized gauge transformation---be
straightforwardly transformed into solutions of the exact Einstein equation. In
cases with nontrivial matter content, the exact stress-energy tensor of the
transformed metric has the same eigenvalues and eigenvectors as the linearized
stress-energy tensor of the initial approximation. When our gauge exists, the
tensorial structure of transformed metric perturbations identically eliminates
all nonlinearities in Einstein's equation. As examples, we derive the exact
Kerr and gravitational plane wave metrics from standard harmonic-gauge
approximations.
