The unsteady dispersion of a solute in a Casson fluid flowing in a conduit (pipe/channel) is studied using the generalized dispersion model of Gill and Sankarasubramanian. With this approach, the entire dispersion process is described appropriately in terms of a simple diffusion process with the effective diffusion coefficient as a function of time, in addition to its dependence on the yield stress of the fluid. The results are accurate up to a first approximation for small times, but verified with Sharp to be exact for large times. The model brings out mainly the effect of yield stress, or equivalently, the plug flow region on the overall dispersion process. It is found that the rate of dispersion is reduced (i.e., the effective diffusivity decreases) due to the yield stress of the fluid, or equivalently, the plug flow region in the conduit. Also, the effective diffusivity increases with time, but eventually attains its steady state value below a critical time [0.48(a2/Dm) for dispersion in a pipe and 0.55(a2/Dm) for dispersion in a channel-the critical transient time for a Newtonian fluid-where "a" is the radius of the pipe and Dm is the molecular diffusivity]. At steady state, for dispersion in a pipe with the plug flow radius one tenth of the radius of the pipe, the effective diffusivity is reduced to about 0.78 times of the corresponding value for a Newtonian fluid at equivalent flow rates; for dispersion in a channel, the reduction factor is about 0.73 confirming the earlier result of Sharp. Further, the location of the center of mass of a passive species over a cross section is found to remain unperturbed during the course of dispersion and for different values of the plug flow parameter (i.e., the yield stress of the fluid). The study can be used as a starting first approximate solution for studying the dispersion in the cardiovascular system or blood oxygenators.