  Observations of OH$^+$ are used to infer the interstellar cosmic ray
ionization rate in diffuse atomic clouds, thereby constraining the propagation
of cosmic rays through and the shielding by interstellar clouds, as well as the
low energy cosmic ray spectrum. In regions where the H$_2$ to H number density
ratio is low, dissociative recombination (DR) is the dominant destruction
process for OH$^+$ and the DR rate coefficient is important for predicting the
OH$^+$ abundance and inferring the cosmic ray ionization rate. We have
experimentally studied DR of electronically and vibrationally relaxed OH$^+$ in
its lowest rotational levels, using an electron--ion merged-beams setup at the
Cryogenic Storage Ring. From these measurements, we have derived a kinetic
temperature rate coefficient applicable to diffuse cloud chemical models, i.e.,
for OH$^+$ in its electronic, vibrational, and rotational ground level. At
typical diffuse cloud temperatures, our kinetic temperature rate coefficient is
a factor of $\sim 5$ times larger than the previous experimentally derived
value and a factor of $\sim 33$ times larger than the value calculated by
theory. Our combined experimental and modelling results point to a significant
increase for the cosmic ray ionization rate inferred from observations of
OH$^+$ and H$_2$O$^+$, corresponding to a geometric mean of $(6.6 \pm 1.0)
\times 10^{-16}\,\mathrm{s}^{-1}$, which is more than a factor of two larger
than the previously inferred values of the cosmic ray ionization rate in
diffuse atomic clouds. Combined with observations of diffuse and dense
molecular clouds, these findings indicate a greater degree of cosmic ray
shielding in interstellar clouds than has been previously inferred.
