In this paper we address the problem of optimal reconstruction of a quantum state from the result of a single measurement, when the original quantum state is known to be a member of some specified set. This process provides both classical information about the state and a reproduction of that state. A suitable figure of merit for this process is the fidelity, which is the probability that the state we construct on the basis of the measurement result is found by a subsequent test to match the original state. We consider the maximization of the fidelity for a mirror symmetric set of three pure qubit states, but find that our results are more generally applicable. In contrast to previous examples, we find that the strategy which minimizes the probability of erroneously identifying the state does not generally maximize the fidelity.