  Let $I$ be any nonempty set and $(M_i, \varphi_i)_{i \in I}$ any family of
nonamenable factors, endowed with arbitrary faithful normal states, that belong
to a large class $\mathcal C_{\rm anti-free}$ of (possibly type III) von
Neumann algebras including all nonprime factors, all nonfull factors and all
factors possessing a Cartan subalgebra. For the free product $(M, \varphi) =
\ast_{i \in I} (M_i, \varphi_i)$, we show that the free product von Neumann
algebra $M$ retains the cardinality $|I|$ and each nonamenable factor $M_i$ up
to stably inner conjugacy, after permutation of the indices. Our main theorem
unifies all previous Kurosh-type rigidity results for free product type II$_1$
factors and is new for free product type III factors. It moreover provides new
rigidity phenomena for type III factors.
