We observe relativistic modifications to the Thomson scattering spectrum in a traditionally classical regime: v(osc)/c = eE(0)/cmomega(0) << 1 and T(e) < 1 keV. The modifications result from scattering off electron-plasma fluctuations with relativistic phase velocities. Normalized phase velocities v/c between 0.03 and 0.12 have been achieved in a N(2) gas-jet plasma by varying the plasma density from 3 x 10(18) cm(-3) to 7 x 10(19) cm(-3) and electron temperature between 85 and 700 eV. For these conditions, the complete temporally resolved Thomson scattering spectrum including the electron and ion features has been measured. A relativistic treatment of the Thomson scattering form factor shows excellent agreement with the experimental data.