  The one-dimensional attractive lattice fermion gas equivalent to the
Heisenberg-Ising spin 1/2 chain is studied for a ring geometry threaded by
magnetic flux. We find that for charged fermions having interaction strength
$\Delta=\cos(\pi / p)$ with $p$ {\it noninteger}, the adiabatic ground state is
periodic in the magnetic flux threading the chain, with period 2 flux quanta,
as found by Shastry and Sutherland for the repulsive case. We find that, at
particular values of the threading field, a sequence of initially zero-energy
bound states form at the Fermi surface during the adiabatic process, the
largest containing $[p]$ (the integer part of $p$) fermions. This largest bound
state moves around to the other Fermi point and sequentially unbinds. We find
Berry's Phase for the whole process to be $[p] \pi$. For $p$ {\it integer}, as
$\Phi$ increases, eventually all the particles in the system go into bound
states of size $[p]$. The period in this case is of order the size of the
system.
