It was shown previously that ultrathin Fe films epitaxially grown on GaAs(001) exhibit a strong in-plane uniaxial magnetic anisotropy which turns out to be a pure interface contribution with an anisotropy constant K-U(S), expressed as an energy per unit area, which is constant in a wide thickness range. However, for films thinner than similar to10 monolayers (ML), K-U(S) decreases with decreasing thickness when measured at 300 K. In order to eliminate effects of thermal excitations, Fe(001) films grown on GaAs(001) by molecular beam epitaxy were investigated by superconducting quantum interference device magnetometry at low temperature. The extrapolated room temperature values and the ground state data both indicate that K-U(S) vanishes at t=2.5 ML. This is the thickness at which the onset of ferromagnetism takes place, i.e., where the Curie temperature T-C becomes nonzero. This might be interpreted as a verification of the Mermin-Wagner theorem in the sense that long-range ferromagnetic order is stabilized by the magnetic anisotropy. It is discussed whether the onset of ferromagnetism is indeed triggered by the appearance of magnetic anisotropy or if there is a common origin of both phenomena. Finally, it is found that the uniaxial anisotropy does not vanish at T-C, but persists up to temperatures of similar to1.5 T-C. This means that K-U(S) does not scale with a certain power of the spontaneous magnetization. The disparity between the persistence of the anisotropy above T-C and its disappearance below the critical thickness of 2.5 ML is discussed. (C) 2002 American Institute of Physics.