Spectra obtained with the Lick 3-m telescope show that a CN-CH band strength anticorrelation exists among a sample of six red giant members of the globular cluster M5 having absolute magnitudes in the range -2.0<M(V)<-1.3. By contrast, a seventh giant, star IV-59 in the color-magnitude survey of Arp (1955, AJ, 60, 317), is found to have bath strong CN and strong CH bands. Carbon and nitrogen abundances determined for five of the observed stars reveal that for the giants exhibiting the CN-CH anticorrelation, (i) carbon is depleted ([C/Fe]<-0.5) by comparison with the [C/Fe] abundances of typical halo subdwarfs, (ii) nitrogen is greatly enhanced (+0.5<[N/Fe]<+1.2) relative to the [N/Fe] abundances of typical subdwarfs, and (iii) the nitrogen and carbon abundances are anticorrelated and correlated, respectively, with the [O/Fe] abundances determined by Sneden et al. (1992, AJ, 104, 2121). These properties are similar to the pattern of CNO abundances reported in the literature for upper-giant-branch stars in other globular clusters such as M92, M3, and M13. By contrast, star M5 IV-59 has [C/Fe] and [N/Fe] abundance ratios that are near the upper Limits for red giants in M5. This star also appears to have an undepleted [O/Fe] abundance, despite a greatly enhanced nitrogen abundance.