A family of solutions to low-energy string theory representing static, spherically symmetric charged black holes is described. They are labeled by their mass, charge, and asymptotic value of the scalar dilaton. The presence of the dilaton is found to have important consequences. In particular, the extremal charged "black holes" are found to be geodesically complete spacetimes with no event horizons and no singularities. Implications of these new solutions for black-hole thermodynamics and open questions in general relativity are also discussed.