We study the quantum-mechanical decay of a Schwarzschild-like black hole into almost-flat space and weak radiation at a very late time, evaluating quantum amplitudes (not just probabilities) for transitions from initial to final states. No information is lost. The model contains gravity and a massless scalar field. The quantum amplitude to go from given initial to final bosonic data in a slightly complexified time-interval $T=τ{\exp}(-iθ)$ at infinity is approximately $\exp(-I)$, where $I$ is the (complex) Euclidean action of the classical solution filling in between the boundary data. And in a locally supersymmetric (supergravity) theory, the amplitude const. exp(-I) is exact. Dirichlet boundary data for gravity and the scalar field are posed on an initial spacelike hypersurface extending to spatial infinity, just prior to collapse, and on a corresponding final spacelike surface, sufficiently far to the future of the initial surface to catch all the Hawking radiation. In an averaged sense this radiation has an approximately spherically-symmetric distribution. If the time-interval $T$ were exactly real, the resulting `hyperbolic Dirichlet boundary-value problem' would not be well posed. If instead (`Euclidean strategy'), one takes $T$ complex, as above ($0

Submitted to Physics Letters B, Friday 23rd July 2004