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 \times 3 bins of richness \lambda and redshift z for \lambda \geq 20 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 _ { 200 m } | \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 , finding M _ { 0 } = [ 3.081 \pm 0.075 ( { stat } ) \pm 0.133 ( { sys } ) ] \cdot 10 ^ { 14 } { M } _ { \odot } at \lambda = 40 and z = 0.35 . The recovered richness scaling index is F = 1.356 \pm 0.051 ( { stat } ) \pm 0.008 ( { sys } ) and the redshift scaling index G = -0.30 \pm 0.30 ( { stat } ) \pm 0.06 ( { sys } ) . These are the tightest measurements of the normalization and richness scaling index made to date from a weak lensing experiment . 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 our 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 .