  Reverberation mapping is a robust method to measure the masses of
supermassive black holes (SMBHs) outside of the local universe. Measurements of
the radius -- luminosity ($R-L$) relation using the Mg II emission line are
critical for determining these masses near the peak of quasar activity at $z
\approx 1 - 2$, and for calibrating secondary mass estimators based on Mg II
that can be applied to large samples with only single-epoch spectroscopy. We
present the first nine Mg II lags from our five-year Australian Dark Energy
Survey (OzDES) reverberation mapping program, which substantially improves the
number and quality of Mg II lag measurements. As the Mg II feature is somewhat
blended with iron emission, we model and subtract both the continuum and iron
contamination from the multi-epoch spectra before analyzing the Mg II line. We
also develop a new method of quantifying correlated spectroscopic calibration
errors based on our numerous, contemporaneous observations of F-stars. The lag
measurements for seven of our nine sources tightly fall on the H$\beta$ $R - L$
relation of \citet{Bentz2013} that has a slope of $\sim0.5$. Our simulations
verify the lag reliability of our nine measurements, and we estimate that the
median false positive rate of the lag measurements is $4\%$.
