  We present constraints on the mean matter density, Omega_m, dark energy
density, Omega_de, and the dark energy equation of state parameter, w, using
Chandra measurements of the X-ray gas mass fraction (fgas) in 42 hot (kT>5keV),
X-ray luminous, dynamically relaxed galaxy clusters spanning the redshift range
0.05<z<1.1. Using only the fgas data for the 6 lowest redshift clusters at
z<0.15, for which dark energy has a negligible effect on the measurements, we
measure Omega_m=0.28+-0.06 (68% confidence, using standard priors on the Hubble
Constant, H_0, and mean baryon density, Omega_bh^2). Analyzing the data for all
42 clusters, employing only weak priors on H_0 and Omega_bh^2, we obtain a
similar result on Omega_m and detect the effects of dark energy on the
distances to the clusters at ~99.99% confidence, with Omega_de=0.86+-0.21 for a
non-flat LCDM model. The detection of dark energy is comparable in significance
to recent SNIa studies and represents strong, independent evidence for cosmic
acceleration. Systematic scatter remains undetected in the fgas data, despite a
weighted mean statistical scatter in the distance measurements of only ~5%. For
a flat cosmology with constant w, we measure Omega_m=0.28+-0.06 and
w=-1.14+-0.31. Combining the fgas data with independent constraints from CMB
and SNIa studies removes the need for priors on Omega_bh^2 and H_0 and leads to
tighter constraints: Omega_m=0.253+-0.021 and w=-0.98+-0.07 for the same
constant-w model. More general analyses in which we relax the assumption of
flatness and/or allow evolution in w remain consistent with the cosmological
constant paradigm. Our analysis includes conservative allowances for systematic
uncertainties. The small systematic scatter and tight constraints bode well for
future dark energy studies using the fgas method. (Abridged)
