Sirius, the seventh-nearest stellar system, is a visual binary containing the metallic-line A1. V star Sirius. A, the brightest star in the sky, orbited in a 50.13. year period by Sirius B, the brightest and nearest white dwarf (WD). Using images obtained over nearly two decades with the Hubble Space Telescope (HST), along with photographic observations covering almost 20 years and nearly 2300 historical measurements dating back to the 19th century, we determine precise orbital elements for the visual binary. Combined with the parallax and the motion of the A component, these elements yield dynamical masses of 2.063 +/- 0.023 M circle dot and 1.018 +/- 0.011 M circle dot for Sirius. A and B, respectively. Our precise HST astrometry rules ou...