Using the NMR F-19 technique, decay of the transverse magnetization (DTM) is studied over the nematic-to-isotropic liquid transition region for aqueous micellar solutions of cesium perfluorononanoate. Transverse magnetization decay appears to be single-component in both nematic and isotropic phases and two-component in the two-phase region. Componental composition of the DTM changes fastly near the low-temperature border of the coexistence region but slowly near the high-temperature border. Long exposition at fixed temperature leads to spatial phase separation of the micellar system : the isotropic phase moves up while the nematic one drowns. It is argued that this fact is due to a difference in micellar concentrations between the two phases.