Carbon isotope ratios, along with carbon and nitrogen abundances, are derived in a sample of 11 red-giant members of one of the most metal-rich clusters in the Milky Way, NGC 6791. The selected red-giants have a mean metallicity and standard deviation of [Fe/H] = +0.39 ± 0.06 (Cunha et al. 2015). We used high-resolution H-band spectra obtained by the SDSS-IV Apache Point Observatory Galactic Evolution Experiment. The advantage of using high-resolution spectra in the H band is that lines of CO are well represented and their line profiles are sensitive to the variation of 12C/13C. Values of the 12C/13C ratio were obtained from a spectrum synthesis analysis. The derived 12C/13C ratios varied between 6.3 and 10.6 in NGC 6791, in agreement with the final isotopic ratios from thermohaline-induced mixing models. The ratios derived here are combined with those obtained for more metal poor red-giants from the literature to examine the correlation between 12C/13C, mass, metallicity, and evolutionary status.