  We study correlation functions in two-dimensional conformal field theory
coupled to induced gravity in the light-cone gauge. Focussing on the fermion
four-point function, we display an unexpected non-perturbative singularity
structure: coupling to gravity {\it qualitatively} changes the perturbative
$(x_1-x_2)^{-1}(x_3-x_4)^{-1}$ singularity into a logarithmic one plus a
non-singular piece. We argue that this is related to the appearence of new
logarithmic operators in the gravitationally dressed operator product
expansions. We also show some evidence that non-conformal but integrable models
may remain integrable when coupled to gravity.
