We present a high-precision mass model of the galaxy cluster MACSJ0416.1-2403, based on a strong-gravitational-lensing analysis of the recently acquired Hubble Space Telescope Frontier Fields (HFF) imaging data. Taking advantage of the unprecedented depth provided by HST/Advanced Camera for Survey observations in three passbands, we identify 51 new multiply imaged galaxies, quadrupling the previous census and bringing the grand total to 68, comprising 194 individual lensed images. Having selected a subset of the 57 most securely identified multiply imaged galaxies, we use the lenstool software package to constrain a lens model comprised of two cluster-scale dark-matter haloes and 98 galaxy-scale haloes. Our best-fitting model predicts image positions with an rms error of 0.68-arcsec, which constitutes an improvement of almost a factor of 2 over previous, pre-HFF models of this cluster. We find the total projected mass inside a 200kpc aperture to be (1.60+/-0.01)x10^14^M_{sun}_, a measurement that offers a three-fold improvement in precision, reaching the per cent level for the first time in any cluster. Finally, we quantify the increase in precision of the derived gravitational magnification of high-redshift galaxies and find an improvement by a factor of ~2.5 in the statistical uncertainty. Our findings impressively confirm that HFF imaging has indeed opened the domain of high-precision mass measurements for massive clusters of galaxies.