  We provide a fairly large family of amalgamated free product groups
$\Gamma=\Gamma_1\ast_{\Sigma}\Gamma_2$ whose amalgam structure can be
completely recognized from their von Neumann algebras. Specifically, assume
that $\Gamma_i$ is a product of two icc non-amenable bi-exact (e.g.,
hyperbolic) groups, and $\Sigma$ is icc amenable and has trivial one-sided
commensurator in $\Gamma_i$, for every $i\in\{1,2\}$. Then $\Gamma$ satisfies
the following rigidity property: any group $\La$ such that $L(\La)$ is
isomorphic to $L(\G)$ admits an amalgamated free product decomposition
$\La=\La_1\ast_\Delta \La_2$ such that the inclusions $L(\Delta)\subseteq
L(\La_i)$ and $L(\Sigma)\subseteq L(\G_i)$ are isomorphic, for every
$i\in\{1,2\}$. This result significantly strengthens some of the previous
Bass-Serre rigidity results for von Neumann algebras. As a corollary, we obtain
the first examples of amalgamated free product groups which are
W$^*$-superrigid.
