  We reconsider the model of neutrino production during the 'bright phase',
first suggested in 1977, in the light of modern understanding of the role of
Pop III stars and acceleration of particles in supernova shocks. We concentrate
on the production of cosmogenic UHE neutrinos in supernova explosions that
accompany the death of Pop III stars. Accelerated protons produce neutrinos in
collisions with CMB photons. We deliberately use simplified assumptions which
make our results transparent. Pop III stars are assumed to be responsible for
the reionization of the universe as observed by WMAP. Since the evolution of
Pop III stars is much faster than the Hubble rate, we consider the burst of UHE
proton production to occur at fixed redshift (z_b=10-20). We discuss the
formation of collisionless shocks and particle acceleration in the early
universe. The composition of accelerated particles is expected to be proton
dominated. A simple calculation is presented to illustrate the fact that the
diffuse neutrinos flux from the bright phase burst is concentrated in a
relatively narrow range around $7.5 \times 10^{15}(20/z_b)^2$ eV. The $\nu_\mu$
flux may be detectable by IceCube without violating the cascade upper limit and
the expected energetics of SNe associated with Pop III stars. A possible
signature of the neutrino production from Pop III stars may be the detection of
resonant neutrino events. For the burst at $z_b=20$ and $\bar{\nu}_e$-flux at
the cascade upper limit, the number of resonant events in IceCube may be as
high as 10 events in 5 years of observations. These events have equal energies,
$E=6.3\times 10^{15}$ eV, in the form of e-m cascades. Given the large
uncertainties in the existing predictions of UHE cosmogenic neutrino fluxes, we
argue that neutrinos from the first stars might become one of the most reliable
hopes for UHE neutrino astronomy.
