  The first stars are known to form in primordial gas, either in minihalos with
about $10^6$~M$_\odot$ or so-called atomic cooling halos of about
$10^8$~M$_\odot$. Simulations have shown that gravitational collapse and disk
formation in primordial gas yield dense stellar clusters. In this paper, we
focus particularly on the formation of protostellar binary systems, and aim to
quantify their properties during the early stage of their evolution. For this
purpose, we combine the smoothed particle hydrodynamics code GRADSPH with the
astrochemistry package KROME. The GRADSPH-KROME framework is employed to
investigate the collapse of primordial clouds in the high-density regime,
exploring the fragmentation process and the formation of binary systems. We
observe a strong dependence of fragmentation on the strength of the turbulent
Mach number $\mathcal{M}$ and the rotational support parameter $\beta{}$.
Rotating clouds show significant fragmentation, and have produced several
Pop.~III proto-binary systems. We report maximum and minimum mass accretion
rates of $2.31 \times 10^{-1}$~M$_{\odot}$ yr$^{-1}$ and $2.18\times
10^{-4}$~M$_{\odot}$ yr$^{-1}$. The mass spectrum of the individual Pop III
proto-binary components ranges from $0.88$~M$_{\odot}$ to $31.96$~M$_{\odot}$
and has a sensitive dependence on the Mach number $\mathcal{M}$ as well as on
the rotational parameter $\beta{}$. We also report a range from $\sim0.01$ to
$\sim1$ for the mass ratio of our proto-binary systems.
