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We have measured the asymmetry in the elastic scattering of electrons from a polarized proton target. An interference between the imaginary part of the two-photon-exchange amplitude and the one-photon-exchange amplitude could produce a polarization effect. The results indicate no asymmetry within the experimental accuracy of 1 to 2% at four-momentum-transfer-squared values values of o.38, 0.59, and 0.98 ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^{2}$.