  Strominger and Vafa have used D-brane technology to identify and precisely
count the degenerate quantum states responsible for the entropy of certain
extremal, BPS-saturated black holes. Here we give a Type-II D-brane description
of a class of extremal {\it and} non-extremal five-dimensional
Reissner-Nordstr\"om solutions and identify a corresponding set of degenerate
D-brane configurations. We use this information to do a string theory
calculation of the entropy, radiation rate and ``Hawking'' temperature. The
results agree perfectly with standard Hawking results for the corresponding
nearly extremal Reissner-Nordstr\"om black holes. Although these calculations
suffer from open-string strong coupling problems, we give some reasons to
believe that they are nonetheless qualitatively reliable. In this optimistic
scenario there would be no ``information loss'' in black hole quantum
evolution.
