  Given a fixed closed manifold M, we exhibit an explicit formula for the
distance function of the canonical L^2 Riemannian metric on the manifold of all
smooth Riemannian metrics on M. Additionally, we examine the (metric)
completion of the manifold of metrics with respect to the L^2 metric and show
that there exists a unique minimal path between any two points. This path is
also given explicitly. As an application of these formulas, we show that the
metric completion of the manifold of metrics is a CAT(0) space.
