  Using a straightforward extension of the analysis of Lieb and Wu, we derive a
simple analytic form for the ground state energy of a one-dimensional Hubbard
ring in the atomic limit. This result is valid for an \textit{arbitrary} number
of lattice sites $L$ and electrons $N \leq L$. Furthermore, our analysis,
including an application of the theory of stochastic matrices, provides insight
into the degeneracy and spin properties of the ground states in the atomic
limit. We give numerical results which illustrate how the atomic limit is
approached.
