  The divergent part of the one-loop off-shell effective action is computed for
a single scalar field coupled to the Ricci curvature of 2D gravity ($c \phi
R$), and self interacting by an arbitrary potential term $V(\phi)$. The
Vilkovisky-DeWitt effective action is used to compute gauge-fixing independent
results. In our background field/covariant gauge we find that the Liouville
theory is finite on shell. Off-shell, we find a large class of renormalizable
potentials which include the Liouville potential. We also find that for
backgrounds satisfying $R=0$, the Liouville theory is finite off shell, as
well.
