The unpolarized, helicity, and transversity distributions of quarks in the proton are calculated in the overlap representation of light-cone wave functions truncated to the lowest order Fock-space components with three valence quarks. The three distributions at the hadronic scale satisfy an interesting relation consistent with the Soffer inequality. Results are derived in a relativistic quark model including evolution up to the next-to-leading order. Predictions for the double transverse-spin asymmetry in Drell-Yan dilepton production initiated by proton-antiproton collisions are presented. Asymmetries of about 20%–30% are found in the kinematic conditions of the PAX experiment.