High resolution echelle spectra of the 8005 Angstrom CN bands are presented for five bright giants in the globular cluster M71 (C1951 + 186, NGC 6838). An analysis of the (CN)-C-13 feature included in this region confirms the results of Briley et al. [ApJ, 429, L119 (1994)]; that the M71 red giants evince low C-12/C-13 ratios and that these ratios are correlated with CN band strengths (CN-strong stars having lower C-12/C-13 ratios). This finding is consistent with the theory that the star-to-star CN band strength variations, as well as the oxygen and possibly sodium differences observed in M71, are related to the CN(O)-cycle of hydrogen burning. (One asymptotic giant branch candidate is also included in the present sample, which exhibits both weak-CN band strength and a low C-12/C-13 ratio.) Examination of published data on CN band strengths and C-12/C-13 ratios among the more metal-poor clusters implies a metallicity dependence in the efficiency of the underlying mechanism responsible for the low C-12/C-13 ratios. A likely candidate appears to be the mixing of CN(O)-processed material into the stellar atmospheres, although a primordial source of CN-processing cannot be ruled out. (C) 1997 American Astronomical Society.