We report the experimental observation of the rotation of the linear polarization of light propagating in a gas of fast-spinning molecules (molecular superrotors). In the observed effect, related to Fermi's prediction of "polarization drag" by a rotating medium, the vector of linear polarization tilts in the direction of molecular rotation. We use an optical centrifuge to bring the molecules in a gas sample to ultrafast unidirectional rotation and measure the polarization drag angles of the order of 10^{-4} rad (with an experimental uncertainty about 10^{-6} rad) over the propagation distance of the order of 1 mm in a number of gases under ambient conditions. We demonstrate an all-optical control of the drag magnitude and direction and investigate the robustness of the mechanical Faraday effect with respect to molecular collisions.