  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. This provides some further evidence for the
veracity of the Auslander Conjecture. Along the way, we provide a simple
cohomological criterion for aspherical manifolds with normal amenable subgroups
of $\pi_1$ to have vanishing simplicial volume. This answers a special case of
a question due to L\"uck.
