This paper presents a model of how hippocampal place cells might be used for spatial navigation in two watermaze tasks: the standard reference memory task and a delayed matching-to-place task. In the reference memory task, the escape platform occupies a single location and rats gradually learn relatively direct paths to the goal over the course of days, in each of which they perform a fixed number of trials. In the delayed matching-to-place task, the escape platform occupies a novel location on each day, and rats gradually acquire one-trial learning, i.e., direct paths on the second trial of each day. The model uses a local, incremental, and statistically efficient connectionist algorithm called temporal difference learning in two distinct components. The first is a reinforcement-based "actor-critic" network that is a general model of classical and instrumental conditioning. In this case, it is applied to navigation, using place cells to provide information about state. By itself, the actor-critic can learn the reference memory task, but this learning is inflexible to changes to the platform location. We argue that one-trial learning in the delayed matching-to-place task demands a goal-independent representation of space. This is provided by the second component of the model: a network that uses temporal difference learning and self-motion information to acquire consistent spatial coordinates in the environment. Each component of the model is necessary at a different stage of the task; the actor-critic provides a way of transferring control to the component that performs best. The model successfully captures gradual acquisition in both tasks, and, in particular, the ultimate development of one-trial learning in the delayed matching-to-place task. Place cells report a form of stable, allocentric information that is well-suited to the various kinds of learning in the model.