We derive abundances of oxygen, sodium, iron-peak, and several traditional "alpha-elements" for giants in the mildly metal-poor globular cluster M5. The data are obtained from Lick Observatory Hamilton Echelle spectra centered on the [O I] doublet. We derive [[Fe/H]] = -1.17+/-0.01 from 13 giants with a range of 350 K in effective temperature. There is no evidence for star-to-star variations in [Fe/H] in this cluster. The range of oxygen abundances is similar to that found earlier in M3 and M92. The largest values of [O/Fe] are near +0.35 and the smallest are near -0.25. The "alpha-elements," e.g., Si, Ca, and Ti, are somewhat overabundant ([el/Fe] approximately +0.2) with little scatter from star to star, as expected on the basis of previous studies of halo field giants in this metallicity domain. The sodium abundances are found to be anticorrelated with the oxygen abundances. Expressed in terms of [O/Fe] and [Na/Fe], the anticorrelation is the same as that found earlier for M3, M13, and the intermediately metal-poor giants of the halo field. Although in M3 and M13 the oxygen and sodium abundances appear to fall into well-separated "O-rich", "O-poor" and "super O-poor" groups, the distribution in M5 is more nearly continuous. We detect no "super O-poor" stars (having [O/Fe] approximately -0.8) in M5. We find that, for those M5 giants in which it has been measured, the strength of the CN band at lambda 3883 is positively correlated with [Na/Fe]. For M13 giants, the CN band strengths are correlated with the abundance of N, and we argue by analogy that the abundance of Na probably is correlated positively with the abundance of N in our sample of M5 giants. This conclusion requires actual nitrogen abundance determinations before it can be substantiated. The anticorrelation of O and Na and the probable correlation of N and Na strongly suggest that there is an ab initio spread in the CNO abundances of M5 stars, and that these elements probably were reshuffled in the cluster by Bethe cycle CNO processing before the stars we now observe were born.