The combined effect of diffusion, and of convection by Poiseuille flow, on the distribution of a small quantity of miscible additive injected into a tube of radius a, is to spread it longitudinally with a Taylor “effective diffusion coefficient”, to an approximation that is good at times greater than about 0.5a2/D (Bailey & Gogarty, 1962), where D is the molecular diffusion coefficient. The present theory, complementary to the Taylor theory, determines the initial action of diffusion on the front of the concentration distribution, to an approximation that is good at times t less than about 0.1a2/D. The theory is exact wherever the added substance does not yet interact with the tube wall, and predicts that the spread in the front due to diffusion extends (Fig. 2) over a distance of order DUt2/a2, where U is the velocity on the axis of the tube. The transition between distributions characteristic of the two theories is illustrated (Fig. 4); and the introduction indicates the relevance of the new theory to work (Caro, 1966) on tracers used in study of the blood circulation.