The kinetics and mechanism of the transformation of vesicular aggregates, prepared from the asymmetric, amphiphilic block copolymer polystyrene-b-poly(acrylic acid) in dioxane-water mixtures, into rod-like structures are investigated using turbidity measurements and transmission electron microscopy. Since the architecture of these aggregates is known to depend upon the solvent composition, small jumps in the dioxane content are used to induce the transformation. The transition proceeds with a conservation of the aggregation number. This process involves two steps, beginning with the quick collapse of the vesicles into a `bowtie' structure. This is followed by a slow rearrangement as each bowtie forms a `dumbbell' aggregate and finally a smooth rod. The relaxation times associated with this morphological change are found to depend upon the initial solvent composition, the magnitude of the dioxane jump, and the initial polymer concentration.