We construct a lattice version of a two-dimensional Liouville field theory. In order to perform Monte Carlo simulations, we introduce a constrained field with vanishing spatial average. A different approach with a coupling to a point source is also considered. We derive RG equations and find a fixed point for our system. We study the relation between observables with and without the constraint on the field, and try to address the issue of a continuum limit. We present some results on the scaling properties of one and two point functions.