Nonradiative energy transfer (NET) was studied in micellar systems of Kraton G-1701 (a diblock copolymer containing polystyrene and hydrogenated polyisoprene blocks) with a polystyrene block tagged with a fluorescence donor (carbazole) and a fluorescence acceptor (anthracene). On addition of selective precipitants for either block of the copolymer, the donor-acceptor pair was located either in the compact micellar core, or in the diffuse solvated shell. It is shown by steady-state and time-resolved fluorescence measurements that the NET is highly efficient in the micellar core and much less efficient in the shell. By measurement of the time-dependent increase in anthracene emission at 413 nm (due to energy transfer, when carbazole is excited at 295 nm) and the concomitant decrease of carbazole emission at 364 nm after mixing two solutions of micelles, labeled by carbazole and anthracene, respectively, the rates of unimer half-arrow-pointing-left-and-right micelle exchange under equilibrium conditions were obtained in several selective precipitants. By comparison with the existing data on the unimer half-arrow-pointing-left-and-right micelle mass exchange rate when relatively large perturbations of the equilibrium conditions are induced, the mass exchange rates under the nonperturbed equilibrium conditions are slower by several orders of magnitude, indicating that a limited segment mobility in micellar cores leads to a slow micelle dissociation.