  We explore the role of the modified Eddington limit due to rapid rotation
(the so-called $\Omega\Gamma-$limit) in the formation of Population III stars.
We performed one-dimensional stellar evolution simulations of mass-accreting
zero-metallicity protostars at a very high rate ($\dot{M} \sim
10^{-3}~\mathrm{M_\odot~yr^{-1}}$) and dealt with stellar rotation as a
separate post-process. The protostar would reach the Keplerian rotation very
soon after the onset of mass accretion, but mass accretion would continue as
stellar angular momentum is transferred outward to the accretion disk by
viscous stress. The protostar envelope expands rapidly when the stellar mass
reaches $5\sim7~\mathrm{M_\odot}$ and the Eddington factor sharply increases.
This makes the protostar rotate critically at a rate that is significantly
below the Keplerian value (i.e., the $\Omega\Gamma-$limit). The resultant
positive gradient of the angluar velocity in the boundary layer between the
protostar and the Keplerian disk prohibits angular momentum transport from the
star to the disk, and consequently further rapid mass accretion. This would
prevent the protostar from growing significantly beyond $20 -
40~\mathrm{M_\odot}$. Another important consequence of the $\Omega\Gamma-$limit
is that the protostar can remain fairly compact ($R \lesssim
50~\mathrm{R_\odot}$) and avoid a fluffy structure ($R \gtrsim
500~\mathrm{R_\odot}$) that is usually found with a very high mass accretion
rate. This effect would make the protostar less prone to binary interactions
during the protostar phase. Although our analysis is based on Pop III protostar
models, this role of the $\Omega\Gamma-$limit would be universal in the
formation process of massive stars, regardless of metallicity.
