We present the third installment of the Yale/San Juan Southern Proper Motion Catalog, SPM3. Absolute proper motions, positions, and photographic BV photometry are given for roughly 10.7 million objects, primarily stars, down to a magnitude of V = 17.5. The catalog covers an irregular area of 3700 deg², between the declinations of -20° and -45°, excluding the Galactic plane. The proper-motion precision for well-measured stars is estimated to be 4.0 mas yr^-1. Unlike previous releases of the SPM Catalog, the proper motions are on the International Celestial Reference System by way of Hipparcos catalog stars and have an estimated systematic uncertainty of 0.4 mas yr^-1. The SPM3 Catalog is available via electronic transfer. As an example of the potential of the SPM3 proper motions, we examine the Galactocentric velocities of a group of metal-poor, main-sequence A stars. The majority of these exhibit thick-disk kinematics, lending support to their interpretation as thick-disk blue stragglers, as opposed to being an accreted component.