  Population III stars, born from the primordial gas in the Universe, lose a
negligible fraction of their mass via stellar winds and possibly follow a
top-heavy mass function. Hence, they have often been regarded as the ideal
progenitors of massive black holes (BHs), even above the pair instability mass
gap. Here, we evolve a large set of Population III binary stars (metallicity
$Z=10^{-11}$) with our population-synthesis code SEVN, and compare them with
Population II binary stars ($Z=10^{-4}$). In our models, the lower edge of the
pair-instability mass gap corresponds to a BH mass of $\approx{86}$
($\approx{91}$) M$_\odot$ for single Population III (II) stars. Overall, we
find only mild differences between the properties of binary BHs (BBHs) born
from Population III and II stars, especially if we adopt the same initial mass
function and initial orbital properties. Most BBH mergers born from Population
III and II stars have primary BH mass below the pair-instability gap, and the
maximum secondary BH mass is $ < 50$ M$_\odot$. Only up to $\approx{3.3}$%
($\approx{0.09}$%) BBH mergers from Population III (II) progenitors have
primary mass above the gap. Unlike metal-rich binary stars, the main formation
channel of BBH mergers from Population III and II stars involves only stable
mass transfer episodes in our fiducial model.
