A water insoluble material, when it is to be moved through a water phase to be delivered into an oil phase, can be conveniently solubilized in a detergent micelle. In that form, it can exist in water at higher concentrations and pass easily into the oil phase for which it has a higher affinity. We have measured the mass transfer rates in water as well as those across the interface. The mass transfer rates in such systems are very low which make them difficult to measure. An experimental setup which is capable of making these measurements is presented. A spatio-temporal variation of concentration is induced inside a spectrophotometer cuvette. As the System moves toward equilibrium under diffusion, the change in concentration with time is monitored by measuring the absorbance of the solution. The solution to the appropriate boundary value problem is used to back calculate the diffusivity and interfacial resistance from the absorbance data using the fact that absorbance is linearly dependent on concentration. Satisfactory values of diffusivities are reported for simple solutes and a surfactant, sodium dodecyl sulfate, SDS (the last has been reported earlier, Neogi, 1994) in water. This technique has been used to measure the transport of nitrobenzene from SDS micelles to squalane. The results indicate significant interfacial resistance. A quantitative model has been provided for the first time.