We measured carbon abundances and the 12C/13C ratio in 31 giant branch stars with previous CN and CH band measurements that span an Mv between -2.33 and 0.18 in the globular cluster M10 (NGC 6254). Abundances were determined by comparing CO features at ∼2.3μm and specifically the 13CO bandhead at 2.37μm, to synthetic spectra generated with MOOG. The observed spectra were obtained with GNIRS on Gemini North with a resolution of R ≈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 12C/13C = 5.10 +/- (+0.18,-0.17) for first generation stars (CN-normal; 13 stars total) and 12C/13C = 4.84 +/- (+0.27,−0.22) for second generation stars (CN-enhanced; 15 stars). We therefore find no statistically significant difference in 12C/13C ratio between stars in either population for the observed magnitude range. Finally, we created models of the expected carbon, nitrogen, and 12C/13C 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 ≈60 to match [C/Fe] abundances, and by a factor of ≈666 to match 12C/13C ratios. We could not simultaneously fit the evolution of both carbon and the 12C/13C ratio with models using the same thermohaline efficiency parameter.