Metastable fcc Fe-Cu alloys in the L1₀ phase have been prepared by stacking one monolayer of Fe and one monolayer of Cu alternately, [Fe(1 ML)/Cu(1 ML)]_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.