We present measurements of the baryon acoustic peak at redshifts z= 0.44, 0.6 and 0.73 in the galaxy correlation function of the final data set of the WiggleZ Dark Energy Survey. We combine our correlation function with lower redshift measurements from the 6-degree Field Galaxy Survey and Sloan Digital Sky Survey, producing a stacked survey correlation function in which the statistical significance of the detection of the baryon acoustic peak is 4.9σ relative to a zero-baryon model with no peak. We fit cosmological models to this combined baryon acoustic oscillation (BAO) data set comprising six distance–redshift data points, and compare the results with similar cosmological fits to the latest compilation of supernovae (SNe) and cosmic microwave background (CMB) data. The BAO and SNe data sets produce consistent measurements of the equation-of-state w of dark energy, when separately combined with the CMB, providing a powerful check for systematic errors in either of these distance probes. Combining all data sets we determine w=−1.03 ± 0.08 for a flat universe, consistent with a cosmological constant model. Assuming dark energy is a cosmological constant and varying the spatial curvature, we find Ω_k=−0.004 ± 0.006.