Measurement of sigma_Total in e+e- Annihilations Below 10.56 GeV

Using the CLEO III detector, we measure absolute cross sections for e+e- -> hadrons at seven center-of-mass energies between 6.964 and 10.538 GeV. R, the ratio of hadronic and muon pair production cross sections, is measured at these energies with a r.m.s. error <2% allowing determinations of the strong coupling alpha_s. Using the expected evolution of alpha_s with energy we find alpha_s(M_Z^2)=0.126 +/- 0.005 ^{+0.015}_{-0.011}, and Lambda=0.31^{+0.09+0.29}_{-0.08-0.21}.