The transmission coefficient between two terminals of an effectively one-dimensional (1D) ring with arbitrary scatterers is calculated exactly as a function of enclosed magnetic flux, φ. 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 φ are found and results are presented in typical cases, for various parameters. The period of the oscillations in the magnetoresistance is φ0 = hc e, though for weak scattering higher harmonics may develop. The oscillations persist even when the elastic scattering is strong. A consequence of this calculation is the inapplicability of the classical addition of resistances in the quantum case. Conditions for the observability of these effects are discussed. © 1984.