Polymerization of rigid rodlike molecules with reactive end groups requires near parallel orientation of the molecules. The reaction is diffusion limited because of the low mobility of the molecules in the later stages of the reaction. Experimental studies have shown that flow-induced molecular orientation enhances the rate of polymerization [Agarwal and Khakhar, Nature 360, 53 (1992)]. Here a theoretical study of the polymerization under axisymmetric extensional flow is carried out to obtain the effective reaction rate constant (k(eff)) for the reaction. Computations show that an increase in the intrinsic rate constant (k(h)) results in a decrease in the relative rate constant k(rel)=k(eff/)k(h). Reduction in the rotational diffusivity (D-r) results in a significant reduction in k(rel); however, the variation of the translational diffusion coefficient perpendicular to the rod axis D-perpendicular to has only a small effect for D-perpendicular to/D(parallel to)much less than1. where D-parallel to is the diffusivity parallel to the rod axis. The imposition of flow increases the effective rate constant, however, the variation of k(rel) with other parameters remains qualitatively similar at different Peclet numbers (Pe= epsilon /D-r, where epsilon is the extensional rate). To simulate the variation of the rate constant during polymerization, computations are carried out for different rod lengths using correlations to estimate rod diffusivities. Results indicate that k(rel) initially decreases and then increases after a certain critical degree of polymerization, which reduces with increase in Peclet number. For sufficiently high extensional rates (epsilon similar to 200 s(-1)) the rate constant becomes higher than the intrinsic value (k(rel)> 1).