  We measured carbon abundances and the $^{12}\mathrm{C}/^{13}\mathrm{C}$ ratio
in 31 giant branch stars with previous CN and CH band measurements that span
-2.33 $<$ M$_{\rm V}$ $<$ 0.18 in the globular cluster M10 (NGC 6254).
Abundances were determined by comparing CO features at $\sim 2.3\, \mu
\mathrm{m}$ and specifically the $^{13}$CO bandhead at $2.37\, \mu \mathrm{m}$,
to synthetic spectra generated with MOOG. The observed spectra were obtained
with GNIRS on Gemini North with a resolution of R $\approx 3500$. The carbon
abundances derived from the IR spectra agree with previous [C/Fe] measurements
found using CN and CH features at the near-UV/blue wavelength range. We found
an average carbon isotope ratio of $^{12}\mathrm{C}/^{13}\mathrm{C}$ =
5.10$_{-0.17}^{+0.18}$ for first generation stars (CN-normal; 13 stars total)
and $^{12}\mathrm{C}/^{13}\mathrm{C}$ = 4.84$_{-0.22}^{+0.27}$ for second
generation stars (CN-enhanced; 15 stars). We therefore find no statistically
significant difference in $^{12}\mathrm{C}/^{13}\mathrm{C}$ ratio between stars
in either population for the observed magnitude range. Finally, we created
models of the expected carbon, nitrogen, and $^{12}\mathrm{C}/^{13}\mathrm{C}$
surface abundance evolution on the red giant branch due to thermohaline mixing
using the MESA stellar evolution code. The efficiency of the thermohaline
mixing must be increased to a factor of $\approx 60$ to match [C/Fe]
abundances, and by a factor of $\approx 666$ to match
$^{12}\mathrm{C}/^{13}\mathrm{C}$ ratios. We could not simultaneously fit the
evolution of both carbon and the $^{12}\mathrm{C}/^{13}\mathrm{C}$ ratio with
models using the same thermohaline efficiency parameter.
