The pathway and kinetics of the order-order transition from shear-aligned cylindrical microdomains to spheres arranged on a bcc lattice in a highly asymmetrical block copolymer are monitored using small-angle neutron scattering (SANS). Shear-aligned hexagonally closed-packed cylindrical microdomains were heated to the spherical microdomain region under quiescent conditions, and the microstructure evolution associated with the order-order transition was followed by SANS. The data indicate a rapid decrease in the cylindrical microdomain order and a gradual rise in the signatures associated with the spherical microdomains, thereby suggesting the existence of an intermediate state where the sample is poorly ordered. The duration of this intermediate state increases with increasing distance from the order-order transition and is consistent with previous rheological measurements on unaligned wormlike cylinders. The results of the scattering measurements are compared to the predictions of recent theories describing the cylinder-to-sphere transition.