The effect of a magnetic flux threading a perfect mesoscopic one-dimensional ring upon the Peierls instability is studied. Within the mean-field approximation, the Peierls transition temperature oscillates with the flux, similar to the superconducting-transition-temperature oscillations in closed geometries, as long as the Peierls transition temperature is less than the level spacing at the Fermi energy. Fluctuations due to the finiteness of the ring, however, destroy this effect as they smear the phase transition. When the opposite effect is considered, i.e., the effect of the Peierls instability upon the oscillatory behavior of thermodynamic quantities of the ring with the flux (e.g., specific heat, persistent current), it is found that the amplitude of the oscillations is suppressed significantly even for a very small ring. In addition, the Peierls transition causes the approximate disappearance of all harmonics of the persistent current except the first one. The last two results are due to the equivalence between the effect of the Peierls gap parameter and that of the temperature in a Peierls insulator.