We consider a growth model built from a weakly asymmetric exclusion process with jump rates depending locally on configurations. In general, invariant measures of the model are unknown. We prove that after rescaling and renormalization this model converges to the Hopf-Cole solution of the KPZ equation driven by Gaussian space-time white noise. This is a generalization of a celebrated result by Bertini and Giacomin for the solid-on-solid model. To avoid the Hopf-Cole transformation and its discrete analogues, we work directly with the KPZ equation using the theory of regularity structures. This is a joint work with Jeremy Quastel from University of Toronto.