  In [BV12] we have proven that, for all hyperbolic groups and for all
non-trivial free products $\Gamma$, the left-right wreath product group
$G:=(Z/2Z)^{(\Gamma)} \rtimes (\Gamma \times \Gamma)$ is W$^*$-superrigid. In
this paper, we extend this result to other classes of countable groups. More
precisely, we prove that for weakly amenable groups $\Gamma$ having positive
first $\ell^2$-Betti number, the same wreath product $G$ is W$^*$-superrigid.
