In situ small-angle X-ray scattering (SAXS) rheology is used to study the dynamic process of shear alignment in a lamellar poly(styrene-b-ethylene propylene) diblock copolymer, at temperatures far below the microphase separation transition temperature. We have focused on the alignment dynamics at time scales shorter than 1 cycle of deformation. To extract this valuable information, we use prealigned specimens and follow subtle changes in the orientation during a steady shear deformation. Two notable changes in the azimuthal SAXS intensity are observed: the maximum of the main peak induced during prealignment shifts to lower azimuthal angles and a secondary peak develops and shifts to higher angles. We effectively modeled the shift of the main peak maximum by assuming that lamellae rotate with the vorticity component of shear. This provides conclusive evidence for lamellar rotation in block copolymers, as opposed to a discontinuous transformation such as lamellar dissolution and reformation. Lamellar rotation also supports our previously proposed mechanism for kink band formation.