The magnetism and its correlation with morphology and structure of ultrathin Fe/Cu(111) films have been studied. At room temperature, the films grow in a quasi-one-dimensional form (stripes) in the submonolayer range. Between 1.4 and 1.8 ML the stripes percolate and become two-dimensional films. The remanent magnetization of the percolated films was observed to be significantly more stable with respect to time than that of the stripes. At low thickness (<2.3 ML) the films adopt the fcc structure from the substrate and later transform to bcc(110) structure with Kurdjumov-Sachs orientation. Experimental evidence suggests that the fcc films have a low-spin ferromagnetic or ferrimagnetic phase, and a perpendicular easy magnetization axis. The magnetization switches to an in-plane high-spin phase after the fcc to bcc structural transformation has been accomplished.