  We prove that every complete non-compact manifold of finite volume contains a
(possibly non-compact) minimal hypersurface of finite volume. The main tool is
the following result of independent interest: if a region $U$ can be swept out
by a family of hypersurfaces of volume at most $V$, then it can be swept out by
a family of mutually disjoint hypersurfaces of volume at most $V +
\varepsilon$.
