  Using the new limit on the neutrino anomalous magnetic moment recently
obtained by the GEMMA experiment on measurements of the cross section for the
reactor antineutrino scattering on free electrons, we get a new direct upper
bound on the neutrino millicharge $\mid q_{\nu} \mid < 1.5 \times 10^{-12}
e_0$. This is a factor of 2 more stringent constraint than the previous bound
obtained from the TEXONO reactor experiment data that is included to the Review
of Particle Properties 2012. We predict that with data from the ongoing new
phase of the GEMMA experiment the upper bound on the neutrino millicharge will
be reduced to $\mid q_{\nu} \mid < 3.7 \times 10^{-13}e_0$ within two years. We
also predict that with the next phase of the considered experiment the upper
bound on the millicharge will be reduced by an order of magnitude over the
present bound and reach the level $\mid q_{\nu} \mid < 1.8 \times 10^{-13}e_0$
within approximately four years.
