Vanishing simplicial volume for certain affine manifolds

We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. As a consequence, these manifolds have zero Euler characteristic, satisfying the Chern Conjecture. Along the way, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups of π1\pi _1 to have vanishing simplicial volume. This answers a special case of a question due to Lück.