  We extract the $e^+e^-\rightarrow \pi^+\pi^-$ 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
$a_\mu^{\pi\pi,\rm LO}(600-900\;\rm MeV) = (368.2 \pm 2.5_{\rm stat} \pm
3.3_{\rm sys})\cdot 10^{-10}$.
