  We present measurements of the total elastic and resonant charge-exchange
ion-atom collision rate coefficient $k_\mathrm{ia}$ of cold sodium (\ce{Na})
with optically-dark low energy \ce{Na+} ions in a hybrid ion-neutral trap. To
determine $k_\mathrm{ia}$, we measured the trap loading and loss from both a
\ce{Na} magneto-optical trap (MOT) and a linear radio frequency quadrupole Paul
trap. We found the total rate coefficient to be $7.4 \pm 1.9 \times 10^{-8}$
cm$^3$/s for the type I \ce{Na} MOT immersed within an $\approx 140$ K ion
cloud and $1.10 \pm 0.25 \times 10^{-7}$ cm$^3$/s for the type II \ce{Na} MOT
within an $\approx 1070$ K ion cloud. Our measurements show excellent agreement
with previously reported theoretical fully quantal \textit{ab initio}
calculations. In the process of determining the total rate coefficient, we
demonstrate that a MOT can be used to probe an optically dark ion cloud's
spatial distribution within a hybrid trap.
