  We prove uniqueness in inverse acoustic scattering in the case the density of
the medium has an unbounded gradient across
$\Sigma\subseteq\Gamma=\partial\Omega$, where $\Omega$ is a bounded open subset
of $\mathbb{R}^{3}$ with a Lipschitz boundary. This follows from a uniqueness
result in inverse scattering for Schr\"odinger operators with singular
$\delta$-type potential supported on the surface $\Gamma$ and of strength
$\alpha\in L^{p}(\Gamma)$, $p>2$.
