Mesoscopic rings with an incommensurate charge density wave (CDW) ground state are investigated theoretically. It is found that the magnetic flux has a pronounced effect on the amplitude of the CDW order parameter. Self-consistent calculations predict a periodic modulation of the order parameter as a function of the flux. In sufficiently small rings the magnetic field can even destroy the CDW ground state. The flux-induced phase transition affects the persistent current and should be observable in the magnetic response.