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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.
