The article presents the effect of non-Newtonian viscosity on the longitudinal dispersion of tracer molecules released in an incompressible viscous non-Newtonian fluid (known as the simplified Cross model fluid) under the action of a constant pressure gradient. The Gill and Sankarasubramanian model is used to solve the unsteady convection diffusion equation for all time periods. An exact expression is obtained for the longitudinal convection coefficient K-1(eta*), which shows the effect of the non-Newtonian parameter * on the centerline coefficient. It is seen that the value of the K-1(eta*) for eta* > 1 is always smaller than the corresponding value for a Newtonian fluid. Also, the longitudinal dispersion coefficient of the solute K-2(tau,eta*) is determined exactly. The results show that the K-2(tau,eta*) asymptotically reaches a stationary state after a certain time. The effect of the eta* on the most dominant dispersion coefficient is clearly depicted. Finally, the axial distribution of the average concentration theta(m) of the solute over the channel cross section is determined at a fixed instant after the solute injection for several values of the eta*. The results for "pure convection" are also reported.