Fe1- xRhx layers are grown with varying rhodium fraction x on (001)-oriented MgO substrates by molecular-beam epitaxy. Film structural, morphological, magnetic, and transport properties are investigated. At room temperature, layers are ferromagnetic (FM) for x < 0.48 and antiferromagnetic (AF) for x > 0.48. Separating the two magnetically ordered phases at x = 0.48 is an abrupt change in the Fe1- xRhx lattice parameter of Δa = 0.0028 nm (Δa/a =-0.9%). For AF layers, the FM state is recovered by heating across a first-order phase transition. The transition leads to a large resistivity modulation, Δρ/ρ = 80%, over a narrow temperature range, ΔT = 3 K, in stoichiometric Fe0.50Rh0.50/MgO(001). For samples with compositions deviating from x = 0.50, fluctuations broaden ΔT and defect scattering reduces Δρ/ρ. © 2018 Author(s).