  We discuss the double-spin asymmetry \aqt in the transversely polarized
Drell-Yan process at small transverse-momentum Q_T of the produced dilepton.
The soft gluon corrections relevant for small Q_T are resummed to all orders in
\alpha_s, up to the next-to-leading logarithmic accuracy. We show that the soft
gluon corrections largely cancel in the spin asymmetry, but the significant
corrections still remain.The asymmetries \aqt are calculated for pp collision
at RHIC and J-PARC, and for p\bar{p} collision at GSI. A novel asymptotic
formula for \aqt at small Q_T is presented, which provides a new approach to
extract the transversity \delta q(x) from the experimental data.
