  The modifications 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 black
hole charge $Q$ and the dilaton mass $m$ satisfies $Q m < O(1)$ then the black
hole has only one horizon. We also argue that for $Q m > O(1)$ 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.
