  We constrain the mass--richness scaling relation of redMaPPer galaxy clusters
identified in the Dark Energy Survey Year 1 data using weak gravitational
lensing. We split clusters into $4\times3$ bins of richness $\lambda$ and
redshift $z$ for $\lambda\geq20$ and $0.2 \leq z \leq 0.65$ and measure the
mean masses of these bins using their stacked weak lensing signal. By modeling
the scaling relation as $\langle M_{\rm 200m}|\lambda,z\rangle = M_0
(\lambda/40)^F ((1+z)/1.35)^G$, we constrain the normalization of the scaling
relation at the 5.0 per cent level as $M_0 = [3.081 \pm 0.075 ({\rm stat}) \pm
0.133 ({\rm sys})] \cdot 10^{14}\ {\rm M}_\odot$ at $\lambda=40$ and $z=0.35$.
The richness scaling index is constrained to be $F=1.356 \pm 0.051\ ({\rm
stat})\pm 0.008\ ({\rm sys})$ and the redshift scaling index $G=-0.30\pm 0.30\
({\rm stat})\pm 0.06\ ({\rm sys})$. These are the tightest measurements of the
normalization and richness scaling index made to date. We use a semi-analytic
covariance matrix to characterize the statistical errors in the recovered weak
lensing profiles. Our analysis accounts for the following sources of systematic
error: shear and photometric redshift errors, cluster miscentering, cluster
member dilution of the source sample, systematic uncertainties in the modeling
of the halo--mass correlation function, halo triaxiality, and projection
effects. We discuss prospects for reducing this systematic error budget, which
dominates the uncertainty on $M_0$. Our result is in excellent agreement with,
but has significantly smaller uncertainties than, previous measurements in the
literature, and augurs well for the power of the DES cluster survey as a tool
for precision cosmology and upcoming galaxy surveys such as LSST, Euclid and
WFIRST.
