  We introduce a new class of groups called wreath-like products. These groups
are close relatives of the classical wreath products and arise naturally in the
context of group theoretic Dehn filling. Unlike ordinary wreath products, many
wreath-like products have Kazhdan's property (T). In this paper, we prove that
any group $G$ in a natural family of wreath-like products with property (T) is
W$^*$-superrigid: the group von Neumann algebra $\text{L}(G)$ remembers the
isomorphism class of $G$. This allows us to provide the first examples (in
fact, $2^{\aleph_0}$ pairwise non-isomorphic examples) of W$^*$-superrigid
groups with property (T).
