We report on a measurement of the asymmetry in the scattering of transversely polarized electrons off unpolarized protons, A( perpendicular), at two Q2 values of 0.106 and 0.230 (GeV/c)(2) and a scattering angle of 30 degrees <theta(e)<40 degrees . The measured transverse asymmetries are A( perpendicular)(Q(2)=0.106 (GeV/c)(2))=(-8.59+/-0.89(stat)+/-0.75(syst))x10(-6) and A( perpendicular)(Q(2)=0.230 (GeV/c)(2))=(-8.52+/-2.31(stat)+/-0.87(syst))x10(-6). The first errors denote the statistical error and the second the systematic uncertainties. From comparison with theoretical estimates of A( perpendicular) we conclude that piN-intermediate states give a substantial contribution to the imaginary part of the two-photon amplitude. There is no obvious reason why this should be different for the real part of the two-photon amplitude, which enters into the radiative corrections for the Rosenbluth separation measurements of the electric form factor of the proton.