We report on a precision measurement of the cross section for the reaction $e^+e^-topi^+pi^-$ in the mass range $0.30<M_{pipi}<1.00$ GeV with the initial state radiation (ISR) method, using 817 pb$^{-1}$ of data at $e^+e^-$ center-of-mass energies near 3.77 GeV and 586 pb$^{-1}$ of data at $e^+e^-$ center-of-mass energies near 4.17 GeV, collected with the CLEO-c detector at the CESR $e^+e^-$ collider at Cornell University. The integrated cross sections in the range $0.30<M_{pipi}<1.00$ GeV for the process $e^+e^-topi^+pi^-$ are determined with a statistical uncertainty of $0.7%$ and a systematic uncertainty of $1.5%$. The leading-order hadronic contribution to the muon anomalous magnetic moment calculated using these measured $e^+e^-topi^+pi^-$ cross sections in the range $M_{pipi}=0.30$ to 1.00 GeV is calculated to be $(500.4pm3.6 (mathrm{stat})pm 7.5(mathrm{syst}))times10^{-10}$.
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