  For any $n\geqslant 4$ let $\tilde B_n=B_n/Z(B_n)$ be the quotient of the
braid group $B_n$ through its center. We prove that any free ergodic
probability measure preserving (pmp) action $\tilde B_n\curvearrowright
(X,\mu)$ is W$^*$-superrigid in the following sense: if
$L^{\infty}(X)\rtimes\tilde B_n\cong L^{\infty}(Y)\rtimes\Lambda$, for an
arbitrary free ergodic pmp action $\Lambda\curvearrowright (Y,\nu)$, then the
actions $\tilde B_n\curvearrowright X,\Lambda\curvearrowright Y$ are stably
(or, virtually) conjugate. Moreover, we prove that the same holds if $\tilde
B_n$ is replaced with a finite index subgroup of the direct product $\tilde
B_{n_1}\times\cdots\times\tilde B_{n_k}$, for some $n_1,\ldots,n_k\geqslant 4$.
The proof uses the dichotomy theorem for normalizers inside crossed products by
free groups from \cite{PV11} in combination with the OE superrigidity theorem
for actions of mapping class groups from \cite{Ki06}.
