  We investigate the formation of dust in a stellar wind during the
red-supergiant (RSG) phase of a very massive Population III star with the
zero-age main sequence mass of 500 M_sun. We show that, in a carbon-rich wind
with a constant velocity, carbon grains can form with a lognormal-like size
distribution, and that all of the carbon available for dust formation finally
condense into dust for wide ranges of the mass-loss rate ((0.1-3)x10^{-3} M_sun
yr^{-1}) and wind velocity (1-100 km s^{-1}). We also find that the
acceleration of the wind driven by newly formed dust suppresses the grain
growth but still allows more than half of gas-phase carbon to be finally locked
up in dust grains. These results indicate that at most 1.7 M_sun of carbon
grains can form in total during the RSG phase of 500 M_sun Population III
stars. Such a high dust yield could place very massive primordial stars as
important sources of dust at the very early epoch of the universe if the
initial mass function of Population III stars was top-heavy. We also briefly
discuss a new formation scenario of carbon-rich ultra-metal-poor stars
considering the feedback from very massive Population III stars.
