The mean square polarizability radii of the proton have been measured for the first time in a virtual-Compton-scattering experiment performed at the MIT-Bates out-of-plane scattering facility. Response functions and polarizabilities obtained from a dispersion analysis of the data at Q2 = 0.057 GeV2/c2 are in agreement with O(p3) heavy baryon chiral perturbation theory. The data support the dominance of mesonic effects in the polarizabilities.