  We use 78 reverberation-measured Mg II time-lag quasars (QSOs) in the
redshift range $0.0033 \leq z \leq 1.89$ to constrain cosmological parameters
in six different cosmological models. The basis of our method is the use of the
radius-luminosity or $R-L$ relation to standardize these 78 Mg II QSOs. In each
cosmological model we simultaneously determine $R-L$ relation and cosmological
model parameters, thus avoiding the circularity problem. We find that the $R-L$
relation parameter values are independent of the cosmological model used in the
analysis thus establishing that current Mg II QSOs are standardizable candles.
Cosmological constraints obtained using these QSOs are significantly weaker
than, but consistent with, those obtained from a joint analysis of baryon
acoustic oscillation (BAO) observations and Hubble parameter [$H(z)$]
measurements. So, we also analyse these QSOs in conjunction with the BAO +
$H(z)$ data and find cosmological constraints consistent with the standard
spatially-flat $\Lambda$CDM model as well as with mild dark energy dynamics and
a little spatial curvature. A larger sample of higher-quality
reverberation-measured QSOs should have a smaller intrinsic dispersion and so
should provide tighter constraints on cosmological parameters.
