The correlation between the broad line region radius and continuum luminosity (R-L relation) of active galactic nuclei (AGNs) is critical for single-epoch mass estimates of supermassive black holes (SMBHs). At z ~ 1-2, where AGN activity peaks, the R-L relation is constrained by the reverberation mapping (RM) lags of the Mg II line. We present 25 Mg II lags from the Australian Dark Energy Survey RM project based on 6 yr of monitoring. We define quantitative criteria to select good lag measurements and verify their reliability with simulations based on both the damped random walk stochastic model and the rescaled, resampled versions of the observed light curves of local, well-measured AGN. Our sample significantly increases the number of Mg II lags and extends the R-L relation to higher redshifts and luminosities. The relative iron line strength $\mathcal {R}_{\rm Fe}$ has little impact on the R-L relation. The best-fitting Mg IIR-L relation has a slope α = 0.39 ± 0.08 with an intrinsic scatter $\sigma _{\rm rl} = 0.15^{+0.03}_{-0.02}$ . The slope is consistent with previous measurements and shallower than the H β R-L relation. The intrinsic scatter of the new R-L relation is substantially smaller than previous studies and comparable to the intrinsic scatter of the H β R-L relation. Our new R-L relation will enable more precise single-epoch mass estimates and SMBH demographic studies at cosmic noon.