  We present the single-epoch black hole mass (M$_{\rm BH}$) calibrations based
on the rest-frame UV and optical measurements of Mg II 2798\AA\ and H$\beta$
4861\AA\ lines and AGN continuum, using a sample of 52 moderate-luminosity AGNs
at z$\sim$0.4 and z$\sim$0.6 with high-quality Keck spectra. We combine this
sample with a large number of luminous AGNs from the Sloan Digital Sky Survey
to increase the dynamic range for a better comparison of UV and optical
velocity and luminosity measurements. With respect to the reference M$_{\rm
BH}$ based on the line dispersion of H$\beta$ and continuum luminosity at
5100\AA, we calibrate the UV and optical mass estimators, by determining the
best-fit values of the coefficients in the mass equation. By investigating
whether the UV estimators show systematic trend with Eddington ratio, FWHM of
H$\beta$, the Fe II strength, and the UV/optical slope, we find no significant
bias except for the slope. By fitting the systematic difference of Mg II-based
and H$\beta$-based masses with the L$_{3000}$/L$_{5100}$ ratio, we provide a
correction term as a function of the spectral index as $\Delta$C = 0.24
(1+$\alpha_{\lambda}$) + 0.17, which can be added to the Mg II-based mass
estimators if the spectral slope can be well determined. The derived UV mass
estimators typically show $>$$\sim$0.2 dex intrinsic scatter with respect to
H$\beta$-based M$_{\rm BH}$, suggesting that the UV-based mass has an
additional uncertainty of $\sim$0.2 dex, even if high quality rest-frame UV
spectra are available.
