We extract the $e^+e^-
ightarrow pi^+pi^-$ cross section in the energy
range between 600 and 900 MeV, exploiting the method of initial state
radiation. A data set with an integrated luminosity of 2.93 fb$^{-1}$ taken at
a center-of-mass energy of 3.773 GeV with the BESIII detector at the BEPCII
collider is used. The cross section is measured with a systematic uncertainty
of 0.9%. We extract the pion form factor $|F_pi|^2$ as well as the
contribution of the measured cross section to the leading order hadronic vacuum
polarization contribution to $(g-2)_mu$. We find this value to be
$a_mu^{pipi,
m LO}(600-900;
m MeV) = (372.5 pm 2.6_{
m stat} pm
3.4_{
m sys})cdot 10^{-10}$.
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