A field of ~38x38 around the supernova remnant (SNR) G349.7+0.2 has been surveyed in the CO J=1-0 transition with the 12 Meter Telescope of the NRAO, using the On-The-Fly technique. The resolution of the observations is 54. We have found that this remnant is interacting with a small CO cloud which, in turn, is part of a much larger molecular complex, which we call the ``Large CO Shell. The Large CO Shell has a diameter of about 100 pc, an H_2 mass of 930,000 solar masses, and a density of 35 cm-3. We investigate the origin of this structure and suggest that an old supernova explosion ocurred about 4 million years ago, as a suitable hypothesis. Analyzing the interaction between G349.7+0.2 and the Large CO Shell, it is possible to determine that the shock front currently driven into the molecular gas is a non-dissociative shock (C-type), in agreement with the presence of OH 1720 MHz masers. The positional and kinematical coincidence among one of the CO clouds that constitute the Large CO Shell, an IRAS point-like source and an ultracompact H II region, indicate the presence of a recently formed star. We suggest that the formation of this star was triggered during the expansion of the Large CO Shell, and suggest the possibility that the same expansion also created the progenitor star of G349.7+0.2. The Large CO Shell would then be one of the few observational examples of supernova-induced star formation.