  Let $M$ be the interior of a connected, oriented, compact manifold $V$ of
dimension at least 2. If each path component of $\partial V$ has amenable
fundamental group, then we prove that the simplicial volume of $M$ is equal to
the relative simplicial volume of $V$ and also to the geometric (Lipschitz)
simplicial volume of any Riemannian metric on $M$ whenever the latter is
finite. As an application we establish the proportionality principle for the
simplicial volume of complete, pinched negatively curved manifolds of finite
volume.
