  We present a non-string-theoretic calculation of the microcanonical entropy
of relic integer-spin Hawking radiation -- at fixed total energy $E$. The only
conserved macroscopic quantity is the total energy $E$ (the total energy of the
relic radiation). Data for a boundary-value approach, with massless,
integer-spin perturbations, are set on initial and final space-like
hypersurfaces. In the resulting 1-dimensional statistical-mechanics problem,
the real part of the (complex) time separation at spatial infinity, $T =
{\mid}T{\mid}\exp(-i\delta), \delta >0$, is the variable conjugate to the total
energy. We count the number of weak-field configurations on the final
space-like hypersurface with energy $E$. One recovers the Cardy formula and the
Bekenstein-Hawking entropy, if Re(T) is of the order of the black-hole life-
time, leading to a statistical interpretation of black-hole entropy. The
microcanonical entropy includes a logarithmic correction to the black-hole area
law, which is {\it universal} (independent of black-hole parameters). Here, the
discreteness of the energy levels is crucial. This approach is compared with
that of string theory for the transition to the fundamental-string r\'egime in
the final stages of evaporation. The squared coupling, $g^2$, regulating the
transition to a highly-excited string state and {\it vice versa}, can be
related to the angle, $\delta$, of complex-time rotation above. The
strong-coupling r\'egime corresponds to a Euclidean black hole, while the
physical limit of a Lorentzian space-time (as $ \delta \to 0_+$) corresponds to
the weak-coupling r\'egime. This resembles the transition to a highly-excited
string-like state which subsequently decays into massless particles, thereby
avoiding the naked singularity.
