First-principles total-energy calculations on tetragonal Fe show that the ferromagnetic and antiferromagnetic phases have tetragonal equilibrium states with c/aϾ1, the fcc value. The bulk layers of an epitaxial film of Fe on Cu͑001͒, which are almost fcc with the Cu lattice constant, are shown to be stable in the antiferromagnetic phase, but inherently unstable in the ferromagnetic phase. The structure of tetragonal equilibrium antiferromagnetic Fe is estimated to be aϭ3.47 Å, cϭ3.75 Å. ͓S0163-1829͑99͒04125-9͔