We present the results of 16 years of monitoring stellar orbits around the massive black hole in the center of the Milky Way, using high-resolution near-infrared techniques. This work refines our previous analysis mainly by greatly improving the definition of the coordinate system, which reaches a long-term astrometric accuracy of ≈300 μas, and by investigating in detail the individual systematic error contributions. The combination of a long-time baseline and the excellent astrometric accuracy of adaptive optics data allows us to determine orbits of 28 stars, including the star S2, which has completed a full revolution since our monitoring began. Our main results are: all stellar orbits are fit extremely well by a single-point-mass potential to within the astrometric uncertainties, which are now ≈6× better than in previous studies. The central object mass is (4.31\, \pm\, 0.06|_\mathrm{stat}\,\pm\, 0.36|_{R_0})\times 10^6 M_\odot, where the fractional statistical error of 1.5% is nearly independent from R 0, and the main uncertainty is due to the uncertainty in R 0. Our current best estimate for the distance to the Galactic center is R 0 = 8.33 ± 0.35 kpc. The dominant errors in this value are systematic. The mass scales with distance as (3.95 ± 0.06) × 106(R 0/8 kpc)2.19 M sun. The orientations of orbital angular momenta for stars in the central arcsecond are random. We identify six of the stars with orbital solutions as late-type stars, and six early-type stars as members of the clockwise-rotating disk system, as was previously proposed. We constrain the extended dark mass enclosed between the pericenter and apocenter of S2 at less than 0.066, at the 99% confidence level, of the mass of Sgr A*. This is two orders of magnitudes larger than what one would expect from other theoretical and observational estimates.