  We show that if $\Gamma = \Gamma_1\times\dotsb\times \Gamma_n$ is a product
of $n\geq 2$ non-elementary ICC hyperbolic groups then any discrete group
$\Lambda$ which is $W^*$-equivalent to $\Gamma$ decomposes as a $k$-fold direct
sum exactly when $k=n$. This gives a group-level strengthening of Ozawa and
Popa's unique prime decomposition theorem by removing all assumptions on the
group $\Lambda$. This result in combination with Margulis' normal subgroup
theorem allows us to give examples of lattices in the same Lie group which do
not generate stably equivalent II$_1$ factors.
