  This paper constructs a Hodge theory of noncompact topologically tame
manifolds $M$. The main result is an isomorphism between the de Rham cohomology
with compact supports of $M$ and the kernel of the Hodge--Witten--Bismut
Laplacian $\lap_\mu$ associated to a measure $d\mu$ which has sufficiently
rapid growth at infinity on $M$. This follows from the construction of a space
of forms associated to $\lap_\mu$ which satisfy an ``extension by zero''
property. The ``extension by zero'' property is proved for manifolds with
cylindrical ends possessing gaussian growth measures.
