  A single-band Hubbard model with nearest and next-nearest neighbour hopping
is studied for $d=1$, 2, 3, using both analytical and numerical techniques. In
one dimension, saturated ferromagnetism is found above a critical value of $U$
for a band structure with two minima and for small and intermediate densities.
This is an extension of a scenario recently proposed by M\"uller--Hartmann. For
three dimensions and non-pathological band structures, it is proven that such a
scenario does not work.
