Stochastic timed automata are an expressive formal model for hard and soft real-time systems. They support choices and delays that can be deterministic, nondeterministic or stochastic. Stochastic choices and delays can be based on arbitrary discrete and continuous distributions. In this paper, we present an analysis approach for stochastic timed automata based on abstraction and probabilistic model checking. It delivers upper/lower bounds on maximum/minimum reachability probabilities and expected cumulative reward values. Based on theory originally developed for stochastic hybrid systems, it is the first fully automated model checking technique for stochastic timed automata. Using an implementation as part of the Modest Toolset and four varied examples, we show that the approach works in practice and present a detailed evaluation of its applicability, its efficiency, and current limitations.