The 1.2 m millimeter-wave telescope at the Center for Astrophysics has been used to carry out the first complete CO survey of M 31. At 115 GHz, the frequency of the J = 1 → 0 transition observed, the telescope has a beamwidth of 8'.7, corresponding to 1.7 kpc at the distance of M31. A total of 8 hr per day during two 6 month observing periods were devoted to the survey, which covers a 153' × 45' area of the disk with half- beamwidth (4'.5) spacing. An average integration time of 100 minutes per point yielded an rms noise of 18 mK in each 1.3 km s-1 wide channel of our spectrometer. The entire galaxy has been observed out to a radius of at least 15 kpc well beyond the optical disk and H I ring.Like other Population I in M3 1, the bulk of the CO emission in M3 1 is confined to a broad ring with a radius of ∼10 kpc. Secondary enhancements in CO near R = 4.5 kpc and 14 kpc, similar to that in other Population I and probably the result of spiral arms, were also detected. The fraction of gas in molecular form drops sharply beyond 10 kpc, but is constant or rising within. Near the tangent of Arp's inner spiral arm N3 at R = 4.5 kpc, the H2/H I surface density ratio is roughly 3 times the mean in the 10 kpc ring. If the Galactic value for the ratio of H2 mass to CO intensity is adopted, the molecular mass surface density at the peak of the 10 kpc ring is ∼1.3 Msun pc-2, about the same as that in the Milky Way at the same radius, but only one-fifth that at the peak of the Milky Way's molecular ring at 4.5 kpc. Even at the brightest CO positions, the fraction of gas in molecular form appears no larger than one-third. The total H2 mass of M31 is found to be 2.7 × 108 Msun, more than a factor of 10 less than that of H I, and about one-fourth the molecular mass of the Milky Way computed in the same way.