We present constraints on the mean matter density, Ωm, dark energy density, ΩDE, and the dark energy equation of state parameter, w, using Chandra measurements of the X-ray gas mass fraction (fgas) in 42 hot (kT > 5 keV), X-ray luminous, dynamically relaxed galaxy clusters spanning the redshift range 0.05 < z < 1.1. Using only the fgas data for the six lowest redshift clusters at z < 0.15, for which dark energy has a negligible effect on the measurements, we measure Ωm= 0.28 ± 0.06 (68 per cent confidence limits, using standard priors on the Hubble constant, H0, and mean baryon density, Ωb h2). Analysing the data for all 42 clusters, employing only weak priors on H0 and Ωb h2, we obtain a similar result on Ωm and a detection of the effects of dark energy on the distances to the clusters at ∼99.99 per cent confidence, with ΩDE= 0.86 ± 0.21 for a non-flat ΛCDM model. The detection of dark energy is comparable in significance to recent type Ia supernovae (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 per cent. For a flat cosmology with a constant dark energy equation of state, we measure Ωm= 0.28 ± 0.06 and w=−1.14 ± 0.31. Combining the fgas data with independent constraints from cosmic microwave background and SNIa studies removes the need for priors on Ωb h2 and H0 and leads to tighter constraints: Ωm= 0.253 ± 0.021 and w=−0.98 ± 0.07 for the same constant-w model. Our most general analysis allows the equation of state to evolve with redshift. Marginalizing over possible transition redshifts 0.05 < zt < 1, the combined fgas+ CMB + SNIa data set constrains the dark energy equation of state at late and early times to be w0=−1.05 ± 0.29 and wet=−0.83 ± 0.46, respectively, in agreement with the cosmological constant paradigm. Relaxing the assumption of flatness weakens the constraints on the equation of state by only a factor of ∼2. Our analysis includes conservative allowances for systematic uncertainties associated with instrument calibration, cluster physics and data modelling. The measured small systematic scatter, tight constraint on Ωm and powerful constraints on dark energy from the fgas data bode well for future dark energy studies using the next generation of powerful X-ray observatories, such as Constellation-X.