  We consider the KPZ equation in one space dimension driven by a stationary
centred space-time random field, which is sufficiently integrable and mixing,
but not necessarily Gaussian. We show that, in the weakly asymmetric regime,
the solution to this equation considered at a suitable large scale and in a
suitable reference frame converges to the Hopf-Cole solution to the KPZ
equation driven by space-time Gaussian white noise. While the limiting process
depends only on the integrated variance of the driving field, the diverging
constants appearing in the definition of the reference frame also depend on
higher order moments.
