  This paper carries out a renormalization of the volume of the
Loewner-Nirenberg singular Yamabe metric in a given conformal class on a
compact manifold-with-boundary. This generalizes the usual volume
renormalization for Poincare-Einstein metrics. The coefficient of the log term
in the volume expansion defines a conformally invariant energy generalizing the
Willmore energy of a surface whose variational derivative with respect to
variations of the boundary hypersurface is a multiple of the obstruction to
smoothness of the singular Yamabe metric itself. The existence of such an
energy answers a question raised by Gover and Waldron.
