It is shown that the tetragonally expanded fcc phase (phase I) of room-temperature-grown Fe films on Cu can be stabilized by hydrogen adsorption up to four monolayers (ML) while the uncovered Fe film starts to transform slightly above 3 ML to the fcc phase (phase II). The changes in the magnetic properties are closely related to the changes in the structure. The phase boundary between the fcc phase II and the bcc phase (phase III) is shifted by 2 ML to lower values upon hydrogen exposure of less than three langmuir after growth at T ≈ 300 K. Growing the Fe films at 300 K under an H₂ atmosphere of ≈ 5x10⁻⁸ mbar results in Fe films, where a spin reorientation transition at about 6 ML is observed. Starting at about 4-5 ML the Fe films transforms into the bcc phase. The structural and magnetic properties of these films are found to he very similar to those observed for low-temperature-grown (100 K) Fe films.