  We study the rate of energy decay for solutions of a coupled wave-heat system
on a rectangular domain. Using techniques from the theory of $C_0$-semigroups,
and in particular a well-known result due to Borichev and Tomilov, we prove
that the energy of classical solutions decays like $t^{-2/3}$ as $t\to\infty$.
This rate is moreover shown to be sharp. Our result implies in particular that
a general estimate in the literature, which predicts at least logarithmic decay
and is known to be best possible in general, is suboptimal in the special case
under consideration here. Our strategy of proof involves direct estimates based
on separation of variables and a refined version of the technique developed in
our earlier paper for a one-dimensional wave-heat system.
