Black holes with a massive dilaton

The modification of dilaton black holes which result when the dilaton acquires a mass are investigated. We derive some general constraints on the number of horizons of the black hole and argue that if the product of the charge $Q$ of the black hole and the dilaton mass $m$ satisfies $\mathrm{Qm}\lesssim 1$ then the black hole has only one horizon. We also argue that for $\mathrm{Qm}\gtrsim 1$ there may exist solutions with three horizons and we discuss the causal structure of such solutions. We also investigate the possible structures of extremal solutions and the related problem of two-dimensional dilaton gravity with a massive dilaton.