Reverberation mapping measurements have been used to constrain the relationship between the size of the broad-line region and luminosity of active galactic nuclei (AGN). This R-L relation is used to estimate single-epoch virial black hole masses, and has been proposed to use to standardize AGN to determine cosmological distances. We present reverberation measurements made with Hβ from the 6-yr Australian Dark Energy Survey (OzDES) Reverberation Mapping Program. We successfully recover reverberation lags for eight AGN at 0.12 < z < 0.71, probing higher redshifts than the bulk of Hβ measurements made to date. Our fit to the R-L relation has a slope of α = 0.41 ± 0.03 and an intrinsic scatter of σ = 0.23 ± 0.02 dex. The results from our multi-object spectroscopic survey are consistent with previous measurements made by dedicated source-by-source campaigns, and with the observed dependence on accretion rate. Future surveys, including LSST, TiDES, and SDSS-V, which will be revisiting some of our observed fields, will be able to build on the results of our first-generation multi-object reverberation mapping survey.