The evolution of a supernova remnant embedded in a strongly magnetized interstellar medium is considered. Two cases can be distinguished: if v(ej)/v(A) is small (v(ej) = initial speed of the ejecta, v(A) = unperturbed Alfven speed), then the magnetic field dictates the evolution of the remnant. The free expansion phase is short and the original energy of the explosion is used up in doing work against the external magnetic field. There is no self-similar deceleration phase. In later stages, the remnant adopts an elongated shape in the direction of the interstellar field and slowly cools off through bremsstrahlung and/or inverse Compton cooling. In the other case, v(ej)2 >> v(A)2, the shock at the outer boundary of the remnant is strong and the magnetic field is not dynamically dominant in the shocked volume. There is an adiabatic deceleration phase which lasts for as long as the post-shock gas pressure is well above the magnetic pressure. When the expansion speed drops to a value approximately v(A), the equatorial expansion is braked, but if v(A) exceeds the external ambient sound speed, expansion continues in the direction of the interstellar field. The paper concludes with a brief discussion of two contexts where these calculations may be relevant: the environment within active galactic nuclei (case v(ej)2/v(A)2 << 1), and the highly magnetized regions close to the centre of our Galaxy (v(ej)2/v(A)2 >> 1).