Virtual Compton scattering on the proton has been investigated at three yet unexplored values of the four-momentum transfer Q^{2}: 0.10, 0.20, and 0.45 GeV^{2}, at the Mainz Microtron. Fits performed using either the low-energy theorem or dispersion relations allowed the extraction of the structure functions P_{LL}-P_{TT}/ε and P_{LT}, as well as the electric and magnetic generalized polarizabilities α_{E1}(Q^{2}) and β_{M1}(Q^{2}). These new results show a smooth and rapid falloff of α_{E1}(Q^{2}), in contrast to previous measurements at Q^{2}=0.33 GeV^{2}, and provide for the first time a precise mapping of β_{M1}(Q^{2}) in the low-Q^{2} region.