Cosmic rays pervade the Galaxy and are thought to be accelerated in supernova shocks. The interaction of cosmic rays with dense interstellar matter has two important effects: 1) high energy (>1 GeV) protons produce {gamma}-rays by {pi}0-meson decay; 2) low energy (< 1 GeV) cosmic rays (protons and electrons) ionize the gas. We present here new observations towards a molecular cloud close to the W51C supernova remnant and associated with a recently discovered TeV {gamma}-ray source. Our observations show that the cloud ionization degree is highly enhanced, implying a cosmic ray ionization rate ~ 10-15 s-1, i.e. 100 times larger than the standard value in molecular clouds. This is consistent with the idea that the cloud is irradiated by an enhanced flux of freshly accelerated low-energy cosmic rays. In addition, the observed high cosmic ray ionization rate leads to an instability in the chemistry of the cloud, which keeps the electron fraction high, ~ 10-5, in a large fraction (Av geq 6mag) of the cloud and low, ~ 10-7, in the interior. The two states have been predicted in the literature as high- and low-ionization phases (HIP and LIP). This is the observational evidence of their simultaneous presence in a cloud.