  We use weak-lensing shear measurements to determine the mean mass of
optically selected galaxy clusters in Dark Energy Survey Science Verification
data. In a blinded analysis, we split the sample of more than 8,000 redMaPPer
clusters into 15 subsets, spanning ranges in the richness parameter $5 \leq
\lambda \leq 180$ and redshift $0.2 \leq z \leq 0.8$, and fit the averaged mass
density contrast profiles with a model that accounts for seven distinct sources
of systematic uncertainty: shear measurement and photometric redshift errors;
cluster-member contamination; miscentering; deviations from the NFW halo
profile; halo triaxiality; and line-of-sight projections. We combine the
inferred cluster masses to estimate the joint scaling relation between mass,
richness and redshift, $\mathcal{M}(\lambda,z) \varpropto M_0 \lambda^{F}
(1+z)^{G}$. We find $M_0 \equiv \langle
M_{200\mathrm{m}}\,|\,\lambda=30,z=0.5\rangle=\left[ 2.35 \pm 0.22\ \rm{(stat)}
\pm 0.12\ \rm{(sys)} \right] \cdot 10^{14}\ M_\odot$, with $F =
1.12\,\pm\,0.20\ \rm{(stat)}\, \pm\, 0.06\ \rm{(sys)}$ and $G = 0.18\,\pm\,
0.75\ \rm{(stat)}\, \pm\, 0.24\ \rm{(sys)}$. The amplitude of the mass-richness
relation is in excellent agreement with the weak-lensing calibration of
redMaPPer clusters in SDSS by Simet et al. (2016) and with the Saro et al.
(2015) calibration based on abundance matching of SPT-detected clusters. Our
results extend the redshift range over which the mass-richness relation of
redMaPPer clusters has been calibrated with weak lensing from $z\leq 0.3$ to
$z\leq0.8$. Calibration uncertainties of shear measurements and photometric
redshift estimates dominate our systematic error budget and require substantial
improvements for forthcoming studies.
