  We derive curvature counterterms in two-dimensional gravity coupled to
conformal matter up to infinite order. By construction the higher-order action
is equivalent to a finite first-order theory with auxiliary scalar. Due to this
equivalence it shares the following remarkable properties: There is no need for
gravitational dressing of the cosmological constant, quantization is consistent
for any conformal anomaly $c$ of the coupled matter system, and if the coupled
matter system is a $c=d~$-dimensional string theory in a Euclidean background
then the effective string theory is $D=d+2~$-dimensional with Minkowski
signature $(1,D-1)$. The resulting quantum theory favours flat geometries and
suppresses both parabolic and hyperbolic singularities.
