  Let $(M, g_0)$ be a closed 4-manifold with positive Yamabe invariant and with
$L^2$-small Weyl curvature tensor. Let $g_1 \in [g_0]$ be any metric in the
conformal class of $g_0$ whose scalar curvature is $L^2$-close to a constant.
We prove that the set of Riemannian metrics in the conformal class $[g_0]$ that
are isospectral to $g_1$ is compact in the $C^\infty$ topology.
