A hyperbolic 3-manifold has finite volume if and only if it is either closed or has toroidal boundary and it is not homeomorphic to $T^2\times I$.

This statement is from [3-Manifold Groups, page 18 (the link is editted)](https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/3-manifold-groups-final-version-031115) by Matthias Aschenbrenner, Stefan Friedl and Henry Wilton, it seems that the three references in the book toward this statement only give partial results (when the boundary components are already cusps). 

**Edits**: The precise statement in my opinion should be:

Let $M$ be a compact three dimensional manifold with incompressible toroidal boundary (possibly none). If the interior of $M$ admits a hyperbolic structure, then $M$ either has finite volume, or is homeomorphic to $T^2\times I$.

Thanks for any solutions or hints.