  The Hubbard model on the kagom\'e lattice has highly degenerate ground states
(the flat lowest band) in the corresponding single-electron problem and
exhibits the so-called flat-band ferromagnetism in the many-electron ground
states as was found by Mielke. Here we study the model obtained by adding extra
hopping terms to the above model. The lowest single-electron band becomes
dispersive, and there is no band gap between the lowest band and the other
band. We prove that, at half-filling of the lowest band, the ground states of
this perturbed model remain saturated ferromagnetic if the lowest band is
nearly flat.
