We report the first clear observations of an entropy-driven phase transition between a dilute micellar "gas" and a disordered but highly associated micellar "liquid" realized with aqueous solutions of poly(ethylene oxide) chains fully end-capped with C-16 and C-18 hydrophobes. Dynamic light scattering and capillary viscometry determine the radii and aggregation numbers of the micelles and, together with the coexisting concentrations, permit estimates of the strength of the entropic attraction through the adhesive hard-sphere model. The behavior is qualitatively consistent with expectations for the entropy gain from the exchange of end blocks between cores of flowerlike micelles consisting of associative triblock copolymers.