We consider a cosmic ray spectrum that is a power law in momentum down to a cutoff, and derive a lower cut-off corresponding to E(kin) approximately (30-60) MeV from the observed ionization rates in nearby diffuse clouds. While the real spectra of cosmic rays may not be so simple, we argue that one expects a substantial change in the spectra of such energies and that, at a first approximation, a power-law spectrum with a lower cut-off is appropriate. Such a description of the cosmic rays in the interstellar medium is not only theoretically more attractive than the spectra used in the literature, but is also supported by recent observations.