We present a high-precision mass model of galaxy cluster Abell 2744, based on a strong gravitational-lensing analysis of the Hubble Space Telescope Frontier Fields (HFF) imaging data, which now include both Advanced Camera for Surveys and Wide Field Camera 3 observations to the final depth. Taking advantage of the unprecedented depth of the visible and near-infrared data, we identify 34 new multiply imaged galaxies, bringing the total to 61, comprising 181 individual lensed images. In the process, we correct previous erroneous identifications and positions of multiple systems in the northern part of the cluster core. With the lenstool software and the new sets of multiple images, we model the cluster using two cluster-scale dark matter haloes plus galaxy-scale haloes for the cluster members. Our best-fitting model predicts image positions with an rms error of 0.79 arcsec, which constitutes an improvement by almost a factor of 2 over previous parametric models of this cluster. We measure the total projected mass inside a 200kpc aperture as (2.162+/-0.005)x10^14^M_{sun}_, thus reaching 1 percent level precision for the second time, following the recent HFF measurement of MACSJ0416.1-2403. Importantly, the higher quality of the mass model translates into an overall improvement by a factor of 4 of the derived magnification factor. Together with our previous HFF gravitational lensing analysis, this work demonstrates that the HFF data enables high-precision mass measurements for massive galaxy clusters and the derivation of robust magnification maps to probe the early Universe.