Faraday rotation, the rotation of the polarization of light due to a magnetic field in the direction of propagation of the light, is used in applications ranging from quantum memory to the detection of biomagnetic fields. For these applications large Faraday rotation is necessary, but absorption of light is detrimental. In search of these properties, we have characterized the Verdet constant of a so far unexplored class of mesogenic organic molecules. We report their spectra and provide an interpretation. A Verdet constant of almost 2.5 × 10^{5} deg T^{-1}m^{-1} is found around 520 nm. This Verdet constant is 3 orders of magnitude larger than the largest known for organic molecules in a region without spectral features. We attribute this enormous Faraday rotation to resonant enhancement by a triplet excitation that does not appear in the linear absorption spectrum and to near-resonant enhancement by low-energy singlet excitations. Furthermore we are able to switch the Faraday rotation by changing the liquid crystal phase of the compound. These results demonstrate a new class of Faraday rotating materials with great potential to replace current materials and improve existing applications. The inherent flexibility in the synthesis of this class of molecules opens a new field of research in Faraday rotation.