  We present next-to-leading order predictions for double transverse-spin
asymmetries in Drell--Yan dilepton production initiated by proton--antiproton
scattering. The kinematic region of the proposed PAX experiment at GSI:
$30\lesssim{s}\lesssim200\GeV^2$ and $2\lesssim{M}\lesssim7\GeV$ is examined.
The Drell--Yan asymmetries turn out to be large, in the range 20--40%.
Measuring these asymmetries would provide the cleanest determination of the
quark transversity distributions.
