Eutectic solidification is very important for exploring new materials in which the periodic multiphase structures may have a remarkable or enhanced functionality. An asymptotic solution of the solute diffusion equation with flow terms for the rod eutectic in the weak convective melt in directional solidification is obtained by using the asymptotic method, and the effect of weak convection on the rod eutectic growth is studied. The so-called weak convection is defined in this paper as the condition in which the intensity of convection flow ahead of the solid liquid interface is relatively small. The relationships between the intensity of convection flow, the growth velocity, the rod spacing and the average interface undercooling can be derived. The result shows that the weak convection has a significant effect on the rod eutectic growth in directional solidification. The average interface undercooling is related to not only the rod spacing and the growth velocity, but also the intensity of convection flow. When specifically focusing on the effect of the intensity of convection flow on the average undercooling in directional solidification, the growth velocity is kept the same. For a certain growth velocity, the average interface undercooling of the rod eutectic decreases as the intensity of convection flow increases, especially at low growth velocity. The rod spacing, which is formed by solidified melt of eutectic or near-eutectic composition, plays a very important role in determining the properties of final products. In this study, by minimizing the average interface undercooling it is found that the rod spacing is a function of growth velocity and the intensity of convection flow. It is shown that for the small growth velocity, the rod spacing increases as the intensity of convection flow increases; for the large growth velocity, the rod spacing increases very slowly as the intensity of convection flow increases. In other words, the smaller the growth velocity, the greater the effect of the weak convection flow on the rod spacing. Our analytical result is compared with the results from other models, and it is also used to calculate the practical case such as the rod spacing of the typical eutectic alloy, Al-Cu eutectic, under the condition of weak forced convection induced by the accelerated crucible rotation technique. It is shown that the rod spacing increases as the rotation rate or the radial position increases, which is consistent with the experimental results obtained by Junze et al.