  A discrete gradient model for interfaces is studied. The interaction
potential is a non-convex perturbation of the quadratic gradient potential.
Based on a representation for the finite volume Gibbs measure obtained via a
renormalization group analysis by Adams, Koteck\'{y} and M\"uller in [AKM] it
is proven that the scaling limit is a continuum massless Gaussian free field.
From probabilistic point of view, this is a Central Limit Theorem for strongly
dependent random fields. Additionally, the convergence of covariances, smoothed
on a scale smaller than the system size, is proven.
