  Let ${\cal D}$ denote the class of bounded real analytic plane domains with
the symmetry of an ellipse. We prove that if $\Omega_1, \Omega_2 \in {\cal D}$
and if the Dirichlet spectra coincide, $Spec(\Omega_1) = Spec(\Omega_2)$, then
$\Omega_1 = \Omega_2$ up to rigid motion.
