The transmission coefficient between two terminals of a one-dimensional ring with arbitrary scatterers is calculated exactly as a function of enclosed magnetic flux. At low temperatures, where the inelastic diffusion length is larger than the size of the ring, its conductance follows from the Landauer formula. Oscillations of the conductance as a function of the characteristics of the scatterers and of (with a period 0=hce) are found. The oscillations persist even when the elastic scattering is strong. © 1984 The American Physical Society.