We investigate the behavior of J-aggregates of a cyanine dye, 3,3(')-di-(gamma-sulfopropyl)-4,5,4('),5(')-dibenzo-9-ethylthiacarbocyanine betaine pyridinium salt, in AOT/water/hexane reverse micelles (W=2-50), both during the course of rapid microemulsion formation and then after stabilization of the microemulsions at thermodynamic equilibrium. We suggest a new mechanism for J-aggregate formation in reverse micelle solutions. According to this model J-aggregates are formed in the aqueous phase prior to formation of a stable microemulsion, rather than in the process of intermicellar exchange. Reverse micelles encapsulate only those J-aggregates whose size is smaller than the mean size of the water pool. Larger J-aggregates located in hexane gradually precipitate from solution. It is demonstrated that micellar solutions can be used as "sieves" for selection and stabilization of J-aggregates with a desired size.