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 $M_0 equiv langle M_{200mathrm{m}},|,lambda=30,z=0.5rangle=left[ 2.35 pm 0.22 rm{(stat)} pm 0.12 rm{(sys)} right] cdot 10^{14} M_odot$, 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 $zleq 0.3$ to $zleq0.8$. Calibration uncertainties of shear measurements and photometric redshift estimates dominate our systematic error budget and require substantial improvements for forthcoming studies.
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