The existence of the long-time tail in the velocity correlation function of a Brownian particle is first discovered from molecular-dynamics simulations and is now well established theoretically and experimentally. In this work, we ask the following question: does this long-time tail have any effect on the kinetics of diffusion-controlled reactions, and if there is any, how the reaction rate is affected, especially in the asymptotic region, t --> infinity? We will show that this long-time tail can be taken into account by the theory developed recently by Dong and Andre. The exact asymptotic solutions to the order of t(-1/2) are found analytically with Smoluchowski and Collins-Kimball boundary conditions. This allows us to reveal chat the long-time tail of the velocity correlation function contributes to the reaction rate an additional term of O(t(-1/2)) to the long-time limit of the classic Smoluchowski and Collins-Kimball theories. (C) 1997 American Institute of Physics. [S0021-9606(97)50347-3].