We present a catalog of extragalactic proper motions created using archival VLBI data and our own VLBA astrometry. The catalog contains 713 proper motions, with average uncertainties of ~24{mu}as/yr, including 40 new or improved proper motion measurements using relative astrometry with the VLBA. The observations were conducted in the X-band and yielded positions with uncertainties of ~70{mu}as. We add 10 new redshifts using spectroscopic observations taken at Apache Point Observatory and Gemini North. With the VLBA Extragalactic Proper Motion Catalog, we detect the secular aberration drift-the apparent motion of extragalactic objects caused by the solar system's acceleration around the Galactic center-at a 6.3{sigma} significance. We model the aberration drift as a spheroidal dipole, with the square root of the power equal to 4.89+/-0.77{mu}as/yr, an amplitude of 1.69+/-0.27{mu}as/yr, and an apex at (275.2{deg}+/-10.0{deg}, -29.4{deg}+/-8.8{deg}). Our dipole model detects the aberration drift at a higher significance than some previous studies, but at a lower amplitude than expected or previously measured. The full aberration drift may be partially removed by the no-net-rotation constraint used when measuring archival extragalactic radio source positions. Like the cosmic microwave background dipole, which is induced by the observer's motion, the aberration drift signal should be subtracted from extragalactic proper motions in order to detect cosmological proper motions, including the Hubble expansion, long-period stochastic gravitational waves, and the collapse of large-scale structure.