In supersymmetric theories the mass of any state is bounded below by the values of some of its charges. The corresponding bounds in the case of Schwarzschild (M>=0) and Reissner-Nordström (M>=||q||) black holes are known to coincide with the requirement that naked singularities be absent. Here we investigate [U(1)]2 charged dilaton black holes in this context. The extreme solutions are shown to 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 supercovariantly constant spinors. The supersymmetric positivity bound for dilaton black holes is given by M>=1/ √2 (||Q||+||P||). This condition for dilaton black holes coincides with the cosmic censorship requirement that the singularities be hidden, as was the case for other asymptotically flat static black-hole solutions. We conjecture that the bounds from supersymmetry and cosmic censorship will coincide for more general solutions as well. The temperature, entropy, and singularity of the stringy black hole are discussed in connection with the extreme limit and restoration of supersymmetry. The Euclidean action (entropy) of the extreme black hole is given by 2π||PQ||. We argue that this result is not altered by higher-order corrections in the supersymmetric theory. In the Lorentzian signature, quantum corrections to the effective on-shell action in the extreme black-hole background are also absent. When a black hole reaches its extreme limit, the thermal description breaks down. It cannot continue to evaporate by emitting (uncharged) 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.