We study a model for quantum gravity on a circle in which the notion of a classical metric tensor is replaced by a quantum metric with an inhomogeneous transformation law under diffeomorphisms. This transformation law corresponds to the co-adjoint action of the Virasoro algebra, and resembles that of the connection in Yang--Mills theory. The transformation property is motivated by the diffeomorphism invariance of the one-dimensional Schrödinger equation. The quantum distance measured by the metric corresponds to the phase of a quantum mechanical wavefunction. The dynamics of the quantum gravity theory are specified by postulating a Riemann metric on the space, Q, of quantum metrics and taking the kinetic energy operator to be the resulting Laplacian on the configuration space . The resulting metric on the configuration space is analysed and found to have singularities. The second-quantized Schrödinger equation is derived, some exact solutions are found, and a generic wavefunction behaviour near one of the metric singularities is described. Finally, some directions for further study are indicated, including an analogue of the Yamabe problem of differential geometry.