It has been well established that Galactic Globular clusters (GCs) harbour more than one stellar population, distinguishable by the anticorrelations of light-element abundances (C-N, Na-O, and Mg-Al). These studies have been extended recently to the asymptotic giant branch (AGB). Here, we investigate the AGB of NGC 6397 for the first time. We have performed an abundance analysis of high-resolution spectra of 47 red giant branch (RGB) and eight AGB stars, deriving Fe, Na, O, Mg, and Al abundances. We find that NGC 6397 shows no evidence of a deficit in Na-rich AGB stars, as reported for some other GCs - the subpopulation ratios of the AGB and RGB in NGC 6397 are identical, within uncertainties. This agrees with expectations from stellar theory. This GC acts as a control for our earlier work on the AGB of M4 (with contrasting results), since the same tools and methods were used.