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 $4times3$ bins of richness $lambda$ and redshift $z$ for $lambdageq20$ 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 $langle M_{rm 200m}|lambda,zrangle = M_0 (lambda/40)^F ((1+z)/1.35)^G$, 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 $G=-0.30pm 0.30 ({rm stat})pm 0.06 ({rm sys})$. 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.
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