We consider the effects of abelian duality transformations on static, spherically symmetric, asymptotically flat string spacetimes in four dimensions, where the dilaton, axion, metric, and gauge fields are allowed to be nonzero. Independent of the α′ expansion, there is a six-parameter family of such configurations, labelled by the charges characterizing the asymptotic behaviour of the various fields: i.e. their mass, dilaton charge, axion charge, electric charge, magnetic charge, and Taub-NUT parameter. We show that duality, based on time-translation invariance, maps these solutions amongst themselves, with the effect of interchanging two pairs of these six labels, namely: (1) the mass and dilaton charge, and (2) the axion charge and the Taub-NUT parameter. We consider in detail the special case of the purely Schwarzschild black hole, for which the mass of the dual configuration vanishes to leading order in α′. Working to next-to-leading order in α′ for the bosonic and heterotic strings, we find that duality takes a black hole of mass M to a (singular) solution having mass ∼ −1/α′M. Finally, we argue that two solutions which are related by duality based on a noncompact symmetry are not always physically equivalent.