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 x 3 bins of richness lambda and redshift z for lambda >= 20 and 0.2 <= z <= 0.65 and measure the mean masses of these bins using their stacked weak lensing signal. By modelling the scaling relation as M-200m|lambda, M-0(lambda/40)(F)((1 + /1.35)(G), we constrain the normalization of the scaling relation at the 5.0 per cent level, finding M-0 = [3.081 +/- 0.075(stat) +/- 0.133(sys)] center dot 10(14) M-aS (TM) at lambda = 40 and = 0.35. The recovered richness scaling index is F = 1.356 +/- 0.051 (stat) +/- 0.008 (sys) and the redshift scaling index G = -0.30 +/- 0.30 (stat) +/- 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 miscentring, cluster member dilution of the source sample, systematic uncertainties in the modelling 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.