  We study the partial transposition ${W}^\Gamma=(\mathrm{id}\otimes
\mathrm{t})W\in M_{dn}(\mathbb C)$ of a Wishart matrix $W\in M_{dn}(\mathbb C)$
of parameters $(dn,dm)$. Our main result is that, with $d\to\infty$, the law of
$m{W}^\Gamma$ is a free difference of free Poisson laws of parameters $m(n\pm
1)/2$. Motivated by questions in quantum information theory, we also derive
necessary and sufficient conditions for these measures to be supported on the
positive half line.
