An experimental investigation of the influence of Brownian motion on shear-induced particle migration of monodisperse suspensions of micron-sized colloidal particles is presented. The suspension is pumped through a 50 micron x 500 micron rectangular cross-section glass channel. The experiments are characterized chiefly by the sample volume fraction (phi = 0.1 - 0.4), and the flow rate expressed as the nondimensional Peclet number (Pe = 10 - 400). For each experiment we measure the entrance length, which is the distance from the inlet of the channel required for the concentration profile to develop to its nonuniform steady state. The entrance length increases strongly with increasing Pe for Pe << 100, in marked contrast to non-Brownian flows for which the entrance length is flow rate independent. For larger Pe the entrance length reaches a constant value which depends on the other experimental parameters. Additionally, the entrance length decreases with increasing phi; this effect is strongest for low phi. Modeling of the migration based on spatial variation of the normal stresses due to the particles captures the primary features observed in the axial evolution over a range of Pe and phi.