  We reconsider the problem of the birefringence of electromagnetic (EM) waves
in a medium consisting of a plasma and a $\nu\bar{\nu}$-gas within the Standard
Model of particle physics. The considered effect arises in such a medium due to
the parity violation for the electroweak neutrino-electron interaction. Our
recent calculations of the electroweak correction to the photon polarization
operator in the electroweak plasma allow us to significantly improve some
previous estimates of such effect in astrophysics. We estimate the rotary power
for EM waves propagating in a non-relativistic plasma in the intergalactic
space and interacting with the gas of relic neutrinos and antineutrinos there.
We show that, in presence of a plasma, the EM wave birefringence effect in a
$\nu\bar{\nu}$-gas exceeds significantly that effect in a $\nu\bar{\nu}$-gas in
empty space considered earlier. These previous treatments of the birefringence
relied on the calculations of the refraction index for on-shell photons in
vacuum using the forward scattering amplitude $\gamma\nu\to \gamma\nu$ with
virtual charged leptons in Feynman diagrams. The possibility to observe
experimentally the new effect suggested here is discussed.
