  Little-Parks oscillation of a M\"obius strip (or ring equivalently) made of a
superconductor is studied based on Ginzburg-Landau theory. It is shown that, if
the strip is wide enough, a novel state appears when the number of magnetic
flux quanta threading the ring is close to a half odd integer. As a result the
shape of Little-Parks oscillation of critical temperature is modified. We
estimate the free energy of this state and give the phase diagram of the
superconducting M\"obius strip in a magnetic field.
