  The two-pion contribution from low energies to the muon magnetic moment
anomaly, although small, has a large relative uncertainty since in this region
the experimental data on the cross sections are neither sufficient nor precise
enough. It is therefore of interest to see whether the precision can be
improved by means of additional theoretical information on the pion
electromagnetic form factor, which controls the leading order contribution. In
the present paper we address this problem by exploiting analyticity and
unitarity of the form factor in a parametrization-free approach that uses the
phase in the elastic region, known with high precision from the Fermi-Watson
theorem and Roy equations for $\pi\pi$ elastic scattering as input. The
formalism also includes experimental measurements on the modulus in the region
0.65-0.70 GeV, taken from the most recent $e^+e^-\to \pi^+\pi^-$ experiments,
and recent measurements of the form factor on the spacelike axis. By combining
the results obtained with inputs from CMD2, SND, BABAR and KLOE, we make the
predictions $a_\mu^{\pi\pi, \LO}\,[2 m_\pi,\, 0.30 \gev]=(0.553 \pm 0.004)
\times 10^{-10}$ and $a_\mu^{\pi\pi, \LO}\,[0.30 \gev,\, 0.63 \gev]=(133. 083
\pm 0.837)\times 10^{-10}$. These are consistent with the other recent
determinations, and have slightly smaller errors.
