M\"ossbauer results obtained at 300 and 4.2 K for ${\mathrm{Fe}}^{57}$ in alloys of 2-40 at.% iron in gold have been fitted with consistent values for the hyperfine interactions. The 300-K data are adequately fitted using a reducing-point-charge model based on a Thomas-Fermi-like model for the iron screening in gold, and the derived parameters are used in fitting the 4.2-K data. The sign of the electric field gradient is deduced to be negative, and information concerning the magnetic moment alignment with different numbers of neighbors is derived. The spins point along $〈111〉$ crystallographic directions when the atom has two or more iron neighbors, and along directions normal to the iron-iron axis when it has one neighbor. With increased size of groups of atoms the spins point along the $〈111〉$ axis that minimizes the number of Fe-Fe axes normal to it, until at about 16 at.% iron long-range ferromagnetism occurs due to the occurrence of linear chains. The results for higher-concentration alloys are consistent with those expected for a ferromagnetic random alloy.