We study an unusual transition between two different types of self-assembled structures in aqueous solutions. Mixtures of a cationic surfactant, CTAB and the organic compound, 5-methyl salicylic acid (5mS) spontaneously self-assemble into unilamellar vesicles at room temperature. Upon heating, these vesicles undergo a thermoreversible transition to "wormlike" micelles, i.e., long, flexible micellar chains. This phase transition results in a 1000-fold increase in the solution viscosity with increasing temperature. Small-angle neutron scattering (SANS) measurements show that the phase transition from vesicles to micelles is a continuous one, with the vesicles and micelles co-existing over a range of temperatures. The tunable vesicle-to-micelle transition and the concomitant viscosity increase upon heating may have utility in a range of areas including microfluidics, drug delivery, and enhanced oil recovery.