We consider the problem of the asymptotic stability of a front type solution to a weakly two dimensional model for the propagation of undular bores. This model includes solutions of the KdV-Burgers equation, so this can be considered a study of the stability of those solutions within a more general two dimensional model. We prove that for domains that are sufficiently narrow in the transverse direction and for a range of the dispersion parameter, the family of fronts is a global asymptotic attractor.