  The stability of a Charge Density Wave (CDW) in a one-dimensional ring
pierced by a Aharonov-Bohm flux is studied in a mean-field picture. It is found
that the stability depends on the parity of the number $N$ of electrons. When
the size of the ring becomes as small as the coherence length $\xi$, the CDW
gap increases for even $N$ and decreases for odd $N$. Then when $N$ is even,
the CDW gap decreases with flux but it increases when $N$ is odd. The variation
of the BCS ratio with size and flux is also calculated. We derive the harmonics
expansion of the persistent current in a presence of a finite gap.
