We present the first application of lens magnification to measure the absolute mass of a galaxy cluster; Abell 1689. The absolute mass of a galaxy cluster can be measured by the gravitational lens magnification of a background galaxy population by the cluster potential. The lensing signal is complicated by the variation in number counts due to galaxy clustering and shot-noise, and by additional uncertainties in relating magnification to mass in the strong lensing regime. Clustering and shot-noise can be dealt with using maximum likelihood methods. Local approximations can then be used to estimate the mass from magnification. Alternatively if the lens is axially symmetric we show that the amplification equation can be solved nonlocally for the surface mass density and the tangential shear. In this paper we present the first maps of the total mass distribution in Abell 1689, measured from the deficit of lensed red galaxies behind the cluster. Although noisier, these reproduce the main features of mass maps made using the shear distortion of background galaxies but have the correct normalisation, finally breaking the ``sheet-mass'' degeneracy that has plagued lensing methods based on shear. We derive the cluster mass profile in the inner 4' (0.48 Mpc/h). These show a profile with a near isothermal surface mass density κ = (0.5+/-0.1)(\theta/1')^{-1} out to a radius of 2.4' (0.28Mpc/h), followed by a sudden drop into noise. We find that the projected mass interior to 0.24 h^{-1}$Mpc is M(<0.24 Mpc/h)=(0.50+/- 0.09) \times 10^{15} Msol/h. We compare our results with masses estimated from X-ray temperatures and line-of-sight velocity dispersions, as well as weak shear and lensing arclets and find all are in fair agreement for Abell 1698. Comment: 14 pages (Latex), 12 postscript figures included, accepted by ApJ