  We study the effects of the first nontrivial hexaquark, $d^*$(2380), on the
equation of state of dense neutron star matter and investigate the consequences
of its existence for neutron stars. The matter in the core regions of neutron
stars is described using density-dependent relativistic mean-field theory. Our
results show that within the parameter spaces examined in our paper, (i) the
critical density at which the $d^*$ condensate emerges lies between 4 and 5
times the nuclear saturation density, (ii) $d^*$ hexaquarks are found to exist
only in rather massive neutron stars, (iii) only relatively small fractions of
the matter in the core of a massive neutron star may contain hexaquarks.
