The magnetic properties of thin epitaxial layers of Fe grown on Cu(111) depend sensitively on the films' structure and morphology. A combination of experiments and numerical simulations reveals that the use of a surfactant monolayer (ML) of Pb during molecular beam epitaxy (MBE) growth at room temperature reduces the amount of interdiffusion at the Cu-Fe interface, retards the fcc-to-bcc transformation by about 2 ML and substantially increases the films' coercivity. The origin of all these alterations to the magnetic behavior can be traced back to the structural modifications provoked by the surfactant during the early growth stages. These results open the way for the controlled fabrication of custom-designed materials with specific magnetic characteristics.