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 out third bodies orbiting either star in the system, down to masses of similar to 15-25 M-Jup. The location of Sirius. B in the Hertzsprung-Russell diagram is in excellent agreement with theoretical cooling tracks for WDs of its dynamical mass, and implies a cooling age of similar to 126 Myr. The position of Sirius. B on the mass-radius plane is also consistent with WD theory, assuming a carbon-oxygen core. Including the pre-WD evolutionary timescale of the assumed progenitor, the total age of Sirius B is about 228 +/- 10 Myr. We calculated evolutionary tracks for stars with the dynamical mass of Sirius A, using two independent codes. We find it necessary to assume a slightly subsolar metallicity, of about 0.85 Z circle dot, to fit its location on the luminosity-radius plane. The age of Sirius. A based on these models is about 237-247. Myr, with uncertainties of +/- 15 Myr, consistent with that of the WD companion. We discuss astrophysical puzzles presented by the Sirius system, including the probability that the two stars must have interacted in the past, even though there is no direct evidence for this and the orbital eccentricity remains high.