  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 they still provide relatively
homogeneous samples of stars that are used to constrain stellar evolution
theory. 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. 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.
