  We present the first simultaneous analysis of the galaxy overdensity and
peculiar velocity fields by modelling their cross-covariance. We apply our new
maximum-likelihood approach to data from the 6-degree Field Galaxy Survey
(6dFGS), which has the largest single collection of peculiar velocities to
date. We present a full derivation of the analytic expression for the
cross-covariance between the galaxy overdensity and peculiar velocity fields
and find direct evidence for a non-zero correlation between the fields on
scales up to $\sim50 h^{-1}$ Mpc. When utilising the cross-covariance, our
measurement of the normalised growth rate of structure is $f\sigma_8(z=0) =
0.424^{+0.067}_{-0.064}$ (15% precision), and our measurement of the
redshift-space distortion parameter is $\beta=0.341^{+0.062}_{-0.058}$ (18%
precision). Both measurements improve by $\sim$20% compared to only using the
auto-covariance information. This is consistent with the literature on
multiple-tracer approaches, as well as Fisher matrix forecasts and previous
analyses of 6dFGS. Our measurement of $f\sigma_8$ is consistent with the
standard cosmological model, and we discuss how our approach can be extended to
test alternative models of gravity.
