  We consider geometries on the space of Riemannian metrics conformally
equivalent to the widely studied Ebin L^2 metric. Among these we characterize a
distinguished metric that can be regarded as a generalization of Calabi's
metric on the space of K\"ahler metrics to the space of Riemannian metrics, and
we study its geometry in detail. Unlike the Ebin metric, the geodesic equation
involves non-local terms, and we solve it explicitly by using a constant of the
motion. We then determine its completion, which gives the first example of a
metric on the space of Riemannian metrics whose completion is strictly smaller
than that of the Ebin metric.
