Astronomical observations suggest that today s Universe is dominated by a dark energy of unknown physical origin. One of the most notable results obtained from many models is that dark energy should cause the expansion of the Universe to accelerate: but the expansion rate as a function of time has proved very difficult to measure directly. We present a new determination of the cosmic expansion history by combining distant supernovae observations with a geometrical analysis of large-scale galaxy clustering within the WiggleZ Dark Energy Survey, using the Alcock-Paczynski test to measure the distortion of standard spheres. Our result constitutes a robust and non-parametric measurement of the Hubble expansion rate as a function of time, which we measure with 10-15 per cent precision in four bins within the redshift range 0.1 <z <0.9. We demonstrate, in a manner insensitive to the assumed cosmological model, that the cosmic expansion is accelerating. Furthermore, we find that this expansion history is consistent with a cosmological-constant dark energy.