M31 was fully surveyed at an angular resolution of 8.'7 in the J = 1 --> 0 rotational transition of CO. The 153' x 45' area observed covers the entire galaxy to a radius of at least 16 kpc, well beyond the optical disk and Population I peak at 10 kpc. The bulk of the CO emission lies between 8 and 11 kpc from the galactic center. Compared with H I, the CO emission is more confined to optical Population I and tends to peak approximately 1 kpc closer to the center. Although previously few CO observations have been directed toward the NE tangent of the Population I ring, it appears as rich in molecular gas as the SW tangent. Little CO emission is found within 8 kpc of the center of M31. Using the Galactic value for the conversion of integrated CO intensity to H-2 mass, the total molecular mass of M31 is approximately 2.5 x 10(8) M., less than one-fourth the molecular mass of the Milky Way computed in the same way.