We consider the transport of a tracer substance within a circular pipe as a result of advection by Poiseuille flow and diffusion. The Péclet number is initially assumed to be infinite, so that molecular diffusion in the axial direction may be neglected. We consider the situation in which the tracer is introduced at an arbitrary point within the pipe, and derive an asymptotic approximation for the concentration as a function of transverse position and time, at a fixed axial position downstream, in the period immediately following the first appearance of tracer. This approximation for the Green's function is axisymmetric at leading order, but for sufficiently small downstream distances the largest correction is found to be an antisymmetric contribution uninfluenced by the wall; further downstream this is supplanted by an axisymmetric wall correction. The resulting approximation is found to describe accurately the early stages of the variation of concentration, in particular the peak commonly observed immediately after the onset of the transient. The combination of the present theory with our previously derived approximation, formally valid at large distances downstream, is found to give reasonably accurate predictions for concentration over the whole course of the transient. Extensions to the case of large but finite Péclet number, and to non‐circular pipes, are briefly considered.