We present results of numerical simulations of the kinetics of exciton-exciton annihilation of weakly localized one-dimensional Frenkel excitons at low temperatures. We find that the kinetics is represented by two well-distinguished components: a fast short-time decay and a very slow long-time tail. The former arises from excitons that initially reside in states belonging to the same localization segment of the chain, while the slow component is caused by excitons created on different localization segments. We show that the usual bimolecular theory fails in the description of the behavior found. We also present a qualitative analytical explanation of the nonexponential behavior observed in both the short- and the long-time decay components. Finally, it is shown that our theoretical estimate for the annihilation time of the fast component is in good agreement with data obtained from transient absorption experiments on J-aggregates of pseudoisocyanine. (C) 2001 American Institute of Physics.