The asymptotic giant branch (AGB) phase is the final stage of nuclear burning for low-mass stars. Although Milky Way globular clusters are now known to harbour (at least) two generations of stars(1,2), they still provide relatively homogeneous samples of stars that are used to constrain stellar evolution theory(3-5). It is predicted by stellar models that the majority of cluster stars with masses around the current turn-off mass (that is, the mass of the stars that are currently leaving the main sequence phase) will evolve through the AGB phase(6,7). Here we report that all of the second-generation stars in the globular cluster NGC 6752-70 per cent of the cluster population-fail to reach the AGB phase. Through spectroscopic abundance measurements, we found that every AGB star in our sample has a low sodium abundance, indicating that they are exclusively first-generation stars. This implies that many clusters cannot reliably be used for star counts to test stellar evolution timescales if the AGB population is included. We have no clear explanation for this observation.