Metastable fcc Fe-Cu alloys in the ${L1}_{0}$ phase have been prepared by stacking one monolayer of Fe and one monolayer of Cu alternately, $[\mathrm{F}\mathrm{e}(1\mathrm{}\mathrm{M}\mathrm{L})/\mathrm{C}\mathrm{u}(1\mathrm{}\mathrm{M}\mathrm{L}){]}_{n}$ (where $n<~25)$ onto a Cu(001) substrate using pulsed-laser deposition. The observed reflection high-energy electron-diffraction oscillations for each Fe and Cu layer and their growth mode as studied by scanning tunneling microscopy indicate a nearly layer-by-layer growth. Alloys up to $n=22$ (22 bilayers) have fcc(001) structure, then change from fcc to bcc structure. Magnetic hysteresis loops measured using the magneto-optical Kerr effect indicate that for $1<~n<~25$ the films are ferromagnetic with the easy axis of magnetization within the plane of the films. For the basic unit of double-layer configuration a Curie temperature of 130 K is estimated. It increases linearly with the number of stackings reaching a saturation value of 400 K at five double layers and beyond.