  The two-pion low-energy contribution to the anomalous magnetic moment of the
muon, $a_\mu\equiv(g-2)_\mu/2$, expres sed as an integral over the modulus
squared of the pion electromagnetic form fac tor, brings a relatively large
contribution to the theoretical error, since the low accuracy of experimental
measurements in this region is amplified by the drastic increase of the
integration kernel. We derive stringent constraints on the two-pion
contribution by exploiting analyticity and unitarity of the pion
electromagnetic form factor. To avoid the poor knowledge of the modulus of this
function, we use instead its phase, known with high precision in the elastic
region from Roy equations for pion-pion scattering via the Fermi-Watson
theorem. Above the inelastic threshold we adopt a conservative integral
condition on the modulus, determined from data and perturbative QCD. Additional
high precision data on the modulus in the range $0.65-0.71$ GeV, obtained from
$e^+e^-$ annihilation and $\tau$-decay experiments, are used to improve the
predictions on the modulus at lower energies by means of a parametrization-free
analytic extrapolation. The results are optimal for a given input and do not
depend on the unknown phase of the form factor above the inelastic threshold.
The present work improves a previous analysis based on the same technique,
including more experimental data and employing better statistical tools for
their treatment. We obtain for the contribution to $a_\mu$ from below 0.63 GeV
the value $(133.258 \pm 0.723)\times 10^{-10}$, which amounts to a reduction of
the theoretical error by about $6 \times 10^{-11}$.
