  We show that for $n \neq 1,4$ the simplicial volume of an inward tame
triangulable open $n$-manifold $M$ with amenable fundamental group at infinity
at each end is finite; moreover, we show that if also $\pi_1(M)$ is amenable,
then the simplicial volume of $M$ vanishes. We show that the same result holds
for finitely-many-ended triangulable manifolds which are simply connected at
infinity.
