  Let $G$ be an arbitrary additive subgroup of $C$ and $Vir[G]$ the
corresponding generalized Virasoro algebra. In the present paper, irreducible
weight modules with finite dimensional weight spaces over $Vir[G]$ are
completely determined. The classification strongly depends on the index group
$G$. If $G$ does not have a direct summand $Z$, then such irreducible modules
over $Vir[G]$ are only modules of intermediate series whose weight spaces are
all 1-dimensional. Otherwise, there is one more class of modules which are
constructed by using intermediate series modules over a generalized Virasoro
subalgebra $Vir[G_0]$ of $Vir[G]$ for a direct summand $G_0$ of $G$ with corank
1.
