  We have made the largest-volume measurement to date of the transition to
large-scale homogeneity in the distribution of galaxies. We use the WiggleZ
survey, a spectroscopic survey of over 200,000 blue galaxies in a cosmic volume
of ~1 (Gpc/h)^3. A new method of defining the 'homogeneity scale' is presented,
which is more robust than methods previously used in the literature, and which
can be easily compared between different surveys. Due to the large cosmic depth
of WiggleZ (up to z=1) we are able to make the first measurement of the
transition to homogeneity over a range of cosmic epochs. The mean number of
galaxies N(<r) in spheres of comoving radius r is proportional to r^3 within
1%, or equivalently the fractal dimension of the sample is within 1% of D_2=3,
at radii larger than 71 \pm 8 Mpc/h at z~0.2, 70 \pm 5 Mpc/h at z~0.4, 81 \pm 5
Mpc/h at z~0.6, and 75 \pm 4 Mpc/h at z~0.8. We demonstrate the robustness of
our results against selection function effects, using a LCDM N-body simulation
and a suite of inhomogeneous fractal distributions. The results are in
excellent agreement with both the LCDM N-body simulation and an analytical LCDM
prediction. We can exclude a fractal distribution with fractal dimension below
D_2=2.97 on scales from ~80 Mpc/h up to the largest scales probed by our
measurement, ~300 Mpc/h, at 99.99% confidence.
