  We study the pion form factor in a broad range of spacelike momentum
transfers within the local-duality version of QCD sum rules. We make use of the
recently calculated two-loop double spectral density of the $<AVA>$ correlator
including O(1) and $O(\alpha_s)$ terms, which allows us to give predictions for
the pion form factor and to study the interplay between the nonperturbative and
perturbative contributions to the pion form factor without any reference to the
pion distribution amplitude. Our results demonstrate the dominance of the
nonperturbative contribution to the form factor up to relatively large values
of the momentum transfer: namely, the nonperturbative O(1) term, which provides
the $1/Q^4$ power correction, gives more than half of the pion form factor in
the region $Q^2 \le 20$ GeV$^2$.
