Very peculiar electronic structure and local magnetic moment distribution for fcc Fe ultrathin films of 1, 2, 3, and 4 monolayers of thickness, epitaxially grown on Cú(001) substrates and covered with some layers of Cu, are calculated. The lattice constant of the Fe film is assumed to be equal to that of Cu, homogeneously in the whole film. The competition between the surface ferromagnetism and bulk antiferromagnetism results in ferromagnetism of 2-monolayer film, asymmetric solution for 3-monolayer film and antisymmetric solution for 4-monolayer film.