We examine the effects of interstellar magnetic fields and cosmic rays on the dynamics of a supernova remnant expanding into a warm H I gas. As long as the shock wave driven by the supernova explosion propagates faster than 110 km s-1, the vicinity of the shock front is fully ionized, and cosmic rays are well coupled to the thermal fluid. They are first accelerated at the adiabatic front, and further compressed in the postshock cooling zone. When the shock velocity drops below 110 km s-1, ion-neutral collisions in the vicinity of the shock dissipate the waves which couple cosmic rays to the thermal gas, and impede cosmic-ray acceleration. We find that magnetic and cosmic-ray pressures together dominate over thermal pressure away from the magnetic poles. As a result, most of the shell becomes considerably thicker, and the shock wave propagates somewhat faster than in the nonmagnetic case. At late times, the transverse mass motions which take place from the poles to the equator create H I holes at the polar caps. Our theory leads to a simple interpretation of the "barrel-shaped" distribution of radio emission observed in some supernova remnants.