  Suppose $G$ is a second countable, locally compact Hausdorff groupoid with
abelian stabilizer subgroups and a Haar system. We provide necessary and
sufficient conditions for the groupoid $C^*$-algebra to have Hausdorff
spectrum. In particular we show that the spectrum of $C^*(G)$ is Hausdorff if
and only if the stabilizers vary continuously with respect to the Fell
topology, the orbit space $G^{(0)}/G$ is Hausdorff, and, given convergent
sequences $\chi_i\to \chi$ and $\gamma_i\cdot\chi_i \to \omega$ in the dual
stabilizer groupoid $\hat{S}$ where the $\gamma_i\in G$ act via conjugation, if
$\chi$ and $\omega$ are elements of the same fiber then $\chi = \omega$
