The document describes the physics case of the PAX experiment using polarized antiprotons, which has recently been proposed for the new Facility for Antiprotons and Ions Research (FAIR) at GSI--Darmstadt. Polarized antiprotons provide access to a wealth of single-- and double--spin observables, thereby opening a new window to physics uniquely accessible at the HESR. The polarized antiprotons would be most efficiently produced by spin--filtering in a dedicated Antiproton Polarizer Ring (APR) using an internal polarized hydrogen gas target. In the proposed collider scenario of the PAX experiment, polarized protons stored in a COSY--like Cooler Storage Ring (CSR) up to momenta of 3.5 GeV/c are bombarded head--on with 15 GeV/c polarized antiprotons stored in the HESR. This asymmetric double--polarized antiproton--proton collider is ideally suited to map, e.g., the transversity distribution in the proton. The proposed detector consists of a large--angle apparatus optimized for the detection of Drell--Yan electron pairs.