  We report results of density-functional theory calculations on the
structural, magnetic, and electronic properties of (1x1)-structures of Co on
Cu(001) for coverages up to two monolayers. In particular we discuss the
tendency towards phase separation in Co islands and the possibility of
segregation of Cu on top of the Co-film. A sandwich structure consisting of a
bilayer Co-film covered by 1ML of Cu is found to be the lowest-energy
configuration. We also discuss a bilayer c(2x2)-alloy which may form due to
kinetic reasons, or be stabilized at strained surface regions. Furthermore, we
study the influence of magnetism on the various structures and, e.g., find that
Co adlayers induce a weak spin-density wave in the copper substrate.
