The structure of epitaxially sputtered single-crystalline Fe/Ni₈₁Fe₁₉(001) multilayers was studied using X-ray diffraction. As the thickness of the Fe layers is increased the structure of the Fe layers changes from vertically expanded fcc(001) to relaxed fcc(001), and finally, to bcc(011), thus identically replicating the structural behavior found in thin Fe films on Cu(001) single crystals. Corresponding to the structural transitions of the Fe layers are changes in the magnetic properties. The relaxed fcc phase contains non-magnetic Fe, which can lead to antiferromagnetic coupling of the Ni₈₁Fe₁₉ layers across the Fe layers.