A spontaneously broken conformal field theory defined on Minkowski spacetime provides a powerful analogy to general relativity. The Goldstone boson associated with the breaking, the dilaton, acts as a scalar graviton. The scalar gravity theory satisfies the equivalence principle and reduces to Newton's laws in the non-relativistic regime. As in general relativity, there is a cosmological constant problem in scalar gravity, and a number of no-go theorems for solutions of the problem in the former case apply to the latter as well. Recently, a mechanism for relaxation of the scalar gravity vacuum energy was proposed which involves a very small violation of the equivalence principle. I will present an explicit realization of the solution using a higher dimensional, weakly coupled dual of the conformal field theory. The setup serves as an interesting analogy to general relativity, but also provides direct implications of theoretical and phenomenological interest.