  In supersymmetric theories the mass of any state is bounded below by the
values of some of its charges. The corresponding bounds in case of
Schwarzschild and Reissner-Nordstr\"om black holes are known to coincide with
the requirement that naked singularities be absent. Here we investigate charged
dilaton black holes in this context. We show that the extreme solutions
saturate the supersymmetry bound of $N=4\ d=4$ supergravity, or dimensionally
reduced superstring theory. Specifically, we have shown that extreme dilaton
black holes, with electric and magnetic charges, admit super-covariantly
constant spinors. The supersymmetric positivity bound for dilaton black holes,
$M \geq \frac{1}{\sqrt 2}(|Q|+|P|)$, takes care of the absence of naked
singularities of the dilaton black holes and is, in this sense, equivalent to
the cosmic censorship condition. The temperature, entropy and singularity are
discussed. The Euclidean action (entropy) of the extreme black hole is given by
$2\pi |PQ|$. We argue that this result, as well as the one for Lorentzian
signature, is not altered by higher order corrections in the supersymmetric
theory. When a black hole reaches its extreme limit, it cannot continue to
evaporate by emitting elementary particles, since this would violate the
supersymmetric positivity bound. We speculate on the possibility that an
extreme black hole may ``evaporate" by emitting smaller extreme black holes.
