We describe a high-resolution, high-sensitivity CO(1-0) survey of a large fraction of the south-west star-forming "ring" 20 kpc in diameter in M31 made with the 14 m FCRAO telescope and its QUARRY receiver array. At the 45'' angular resolution of the telescope, the ring is resolved into giant complexes of molecular gas with typical sizes of a few hundred parsecs and masses in excess to 106 M☉. Most of these delineate a prominent spiral arm coincident with that traced by atomic hydrogen and dust. This arm appears in all respects quite similar to a prominent spiral arm in our Galaxy at about the same Galactic radius: the Carina arm. Closer to the center, a second fainter molecular spiral arm segment with an H I and dust counterpart is also detected. The main OB associations and the UV light define an arm offset from that traced by the gas by about 500 pc. The H II regions are mainly found either along the edge of the prominent gaseous arm which faces the associations, or aligned with the second fainter gaseous arm. They define Baade's arms S4 and S3, respectively.