  I calculate the evolution of Be and B abundances produced by cosmic rays
generated by massive stars in the pregalactic phase of the universe. The inputs
for calculation, i.e. the star formation rate and the nuclear abundances of
cosmic rays, which I assume to be the same as those of the ISM, are taken from
the results of a detailed cosmic chemical evolution model with its parameters
best fitted from several items of observational information including an early
reionization of the IGM by $z\sim 15$. I found that when the $^6$Li plateau
abundance observed in metal-poor halo stars originated in the pregalactic
cosmological cosmic ray nucleosynthesis, Be and B simultaneously produced with
$^6$Li amount to the lowest levels ever detected in metal-poor halo stars. It
is desirable to observe Be and B abundances in metal-poor halo stars with
[Fe/H]$\leq -3$ in order to elucidate the possibility of early $^6$LiBeB
production by pregalactic supernova cosmic ray nucleosynthesis.
