We present stellar proper motions in the Galactic bulge from the Sagittarius Window Eclipsing Extrasolar Search (SWEEPS) project using ACS WFC on HST. Proper motions are extracted for more than 180,000 objects, with >81,000 measured to accuracy better than 0.3 mas yr⁻¹ in both coordinates. We report several results based on these measurements: (1) Kinematic separation of bulge from disk allows a sample of >15,000 bulge objects to be extracted based on ≥6 σ detections of proper motion, with <0.2% contamination from the disk. This includes the first detection of a candidate bulge blue straggler population. (2) Armed with a photometric distance modulus on a star-by-star basis, and using the large number of stars with high-quality proper-motion measurements to overcome intrinsic scatter, we dissect the kinematic properties of the bulge as a function of distance along the line of sight. This allows us to extract the stellar circular speed curve from proper motions alone, which we compare with the circular speed curve obtained from radial velocities. (3) We trace the variation of the {l, b} velocity ellipse as a function of depth. (4) Finally, we use the density-weighted {l, b} proper-motion ellipse produced from the tracer stars to assess the kinematic membership of the 16 transiting planet candidates discovered in the Sagittarius Window; the kinematic distribution of the planet candidates is consistent with that of the disk and bulge stellar populations.