  We study a one-dimensional model of interacting conduction electrons with a
two-fold degenerate band away from half filling. The interaction includes an
on-site Coulomb repulsion and Hund's rule coupling. We show that such
one-dimensional system has a divergent Cooper pair susceptibility at T = 0,
provided the Coulomb interaction $U$ between electrons on the same orbital and
the modulus of the Hund's exchange integral $|J|$ are larger than the
interorbital Coulomb interaction. It is remarkable that the superconductivity
can be achieved for {\it any} sign of $J$. The opening of spectral gaps makes
this state stable with respect to direct electron hopping between the orbitals.
The scaling dimension of the superconducting order parameter is found to be
between 1/4 (small $U$) and 1/2 (large $U$).
