New H-alpha and [S II] images of the HH 111 jet taken with the Hubble Space Telescope reveal marked proper motions and morphological changes when compared with similar images obtained four years earlier. Knots in the jet, which are dominated by emission from nested bow shocks, generally move ballistically, with no evidence for turbulent motions even in regions where the emission has a complex morphology. These bow shocks sometimes overtake one another; the new images show this occurred in knot L about 80 years ago. Photometric variability, clearly visible for the first time at subarcsecond scales, can confuse ground-based measurements that require many years between epochs to detect reliable proper motions. With the exception of the bow shock L, whose wings expand laterally, the jet moves mainly along its long axis. Because HH 111 lies nearly in the plane of the sky, the proper motions translate accurately to space velocities, which range from 220 km/s to 330 km/s with a typical uncertainty of +/- 5 km/s . The fastest knots are associated with object E at the base of the visible jet, where a cooling layer is in the process of forming behind one of the shocks. Velocity differences between adjacent knots within the optically bright part of the jet are typically 40 km/s, in line with predictions of nonmagnetic shock models based on emission line fluxes. This agreement limits the component of the magnetic field perpendicular to the axis of the jet to be <~ 1 mG.