We present observations of CO J = 1-0 emission toward 3C 391, a supernova remnant whose radio continuum and X-ray emission suggest evolution near a strong density gradient in the surrounding medium. The CO maps reveal that the remnant is located at the edge of a molecular cloud, and the CO emission shows a striking correspondence with the ``breakout'' morphology of the remnant traced by radio continuum emission. These data support the idea that the progenitor star exploded within a dense molecular cloud and that the supernova blast wave has now broken out through the cloud boundary. Bright radio emission coincident with strong CO emission suggests that the blast wave accelerates electrons to energies of at least 20 GeV, even as it moves into dense neutral material.