  We use the recently measured accurate BaBaR data on the modulus of the pion
electromagnetic form factor, $F_\pi(t)$, up to an energy of 3 GeV, the I=1
$P$-wave phase of the $\pi\pi$ scattering amplitude up to the $\omega-\pi$
threshold, the pion charge radius known from Chiral Perturbation Theory, and
the recently measured JLAB value of $F_\pi$ in the spacelike region at $t=-2.45
{\rm GeV}^2$ as inputs in a formalism that leads to bounds on $F_\pi$ in the
intermediate spacelike region. We compare our constraints with experimental
data and with perturbative QCD along with the results of several theoretical
models for the non-perturbative contributions proposed in the literature.
