  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 $T={\tau}{\exp}(-i{\theta})$ 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
($0<\theta{\leq}{\pi}/2$), the field equations become strongly elliptic, with a
unique solution to the classical boundary-value problem. Expanding the bosonic
part of the action to quadratic order in perturbations about the classical
solution gives the quantum amplitude for weak-field final configurations, up to
normalization. Such amplitudes are calculated for weak final scalar fields.
