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We have measured the parity-violating electroweak asymmetry in the elastic scattering of polarized electrons from 12C nuclei. Our result is Aexpt=0.60±0.14±0.02 ppm, where the first error is statistical and the second is systematic. With a beam polarization of 0.37, we compute the isoscalar vector hadronic coupling constant γ̃ to be 0.136±0.032±0.009. The standard model predicts γ̃ =0.155 at the tree level, in agreement with our data.