  Let $\mu$ be a positive finite measure on the unit circle and $\mathcal{D}
(\mu)$ the associated Dirichlet space. The generalized Brown-Shields conjecture
asserts that an outer function $f \in \mathcal{D} (\mu )$ is cyclic if and only
if $c\_\mu (Z (f))= 0$, where $c\_\mu$ is the capacity associated with
$\mathcal{D} (\mu)$ and $Z(f)$ is the zero set of $f$. In this paper we prove
that this conjecture is true for measures with countable support. We also give
in this case a complete and explicit characterization of invariant subspaces.
