We present time-resolved small-angle X-ray scattering (SAXS) measurements of the kinetics of a barotropic disorder-order transition in polystyrene-polyisoprene (SI) diblock micelles in diethyl phthalate (DEP), a styrene selective solvent. Because of the fast equilibration in a pressure jump we were able to observe the initial induction stage during which the short-range order of the micellar fluid increases, followed by a sharp first-order nucleation and growth of the BCC phase and eventual late stage coarsening. The time evolution of intensity from individual diffraction spots of a crystallite was obtained from the analysis of 2-dimensional SAXS patterns and shows late stage power-law growth. The time evolution of the SAXS intensity profiles was analyzed by fitting the contribution to the scattering from the ordered Bragg peaks by Gaussians and analyzing the contribution to the scattering from the disordered micellar liquid in terms of the Percus-Yevick interacting hard sphere model The time evolution of the micellar structural parameters obtained from this analysis reveals that the disorder to order transition occurs when the volume fraction of micelles reaches 0.53, and the effective hard sphere radius and volume fraction of micelles increases upon ordering while the core radius is unaffected. In both the ordered and disordered states the hard sphere radius and volume fraction of micelles increase with increasing pressure indicating increased swelling of the micellar corona. In the ordered state there is considerable penetration of the corona chains from neighboring micelles.