Previous studies of dilute ZnFe alloys have indicated that, although the iron atom does not possess a well defined localised magnetic moment, it is on the verge of magnetism. In this work alloy samples containing up to 0.4 at.% Fe in Zn, prepared by a rapid quenching technique to overcome the limited solubility, have been studied using the Mossbauer effect in 57Fe. The zinc-rich intermetallic compound Zn13Fe has also been similarly studied. Measurements at temperatures down to 4.2K and in applied magnetic fields up to 60 kG show that the mean square displacements of the Fe atoms in Zn are markedly anisotropic, as expected from elasticity measurements and lattice dynamic calculations. In the alloys and in Zn13Fe the electric quadrupole splitting is shown to be virtually temperature-independent. The high-field-low-temperature spectra, although complex, are shown to indicate a very low limit to the local susceptibility consistent with the conclusion that the Fe solute atoms in Zn carry no appreciable local moment.