Molecular clouds interacting with supernova remnants may be subject to a greatly enhanced irradiation by cosmic rays produced at the shocked interface between the ejecta and the molecular gas. Over the past decade, broad-band observations have provided important clues about these relativistic particles and indicate that they may dominate over the locally observed cosmic-ray population by a significant amount. In this paper, we estimate the enhancement and find that the cosmic ray energy density can be up to $\sim$1000 times larger in the molecular cloud than in the field. This enhancement can last for a few Myr and leads to a corresponding increase in the ionization fraction, which has important consequences for star formation. Ionization f...