We report on work to increase the number of well-measured Type Ia supernovae (SNe Ia) at high redshifts. Light curves, including high signal-to-noise Hubble Space Telescope data, and spectra of six SNe Ia that were discovered during 2001, are presented. Additionally, for the two SNe with z > 1, we present ground-based J-band photometry from Gemini and the Very Large Telescope. These are among the most distant SNe Ia for which ground-based near-IR observations have been obtained. We add these six SNe Ia together with other data sets that have recently become available in the literature to the Union compilation. We have made a number of refinements to the Union analysis chain, the most important ones being the refitting of all light curves with the SALT2 fitter and an improved handling of systematic errors. We call this new compilation, consisting of 557 SNe, the Union2 compilation. The flat concordance ΛCDM model remains an excellent fit to the Union2 data with the best-fit constant equation-of-state parameter w = -0.997+0.050 -0.054(stat)+0.077 -0.082(stat + sys together) for a flat universe, or w = -1.038+0.056 -0.059(stat)+0.093 -0.097(stat + sys together) with curvature. We also present improved constraints on w(z). While no significant change in w with redshift is detected, there is still considerable room for evolution in w. The strength of the constraints depends strongly on redshift. In particular, at z >~ 1, the existence and nature of dark energy are only weakly constrained by the data. Based in part on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute (STScI). STScI is operated by the Association of Universities for Research in Astronomy (AURA), Inc. under the NASA contract NAS 5-26555. The observations are associated with programs HST-GO-08585 and HST-GO-09075. Based, in part, on observations obtained at the ESO La Silla Paranal Observatory (ESO programs 67.A-0361 and 169.A-0382). Based, in part, on observations obtained at the Cerro-Tololo Inter-American Observatory (CTIO), National Optical Astronomy Observatory (NOAO). Based on observations obtained at the Canada-France-Hawaii Telescope (CFHT). Based, in part, on observations obtained at the Gemini Observatory (Gemini programs GN-2001A-SV-19 and GN-2002A-Q-31). Based, in part on observations obtained at the Subaru Telescope. Based, in part, on data that were obtained at the W. M. Keck Observatory.