  We study theoretically diffusion of one-dimensional Frenkel excitons in
J-aggregates at temperatures that are smaller or of the order of the J-band
width. We consider an aggregate as an open linear chain with uncorrelated
on-site (diagonal) disorder that localizes the exciton at chain segments of
size smaller than the full chain length. The exciton diffusion over the
localization segments is considered as incoherent hopping. The diffusion is
probed by the exciton fluorescence quenching which is due to the presence of
point traps in the aggregate. The rate equation for populations of the
localized exciton states is used to describe the exciton diffusion and
trapping. We show that there exist two regimes of the exciton diffusion at low
temperatures. The first, slower one, involves only the states of the very tail
of the density of states, while the second, much faster one, also involves the
higher states that are close to the bottom of the exciton band. The activation
energy for the first regime of diffusion is of the order of one fifth of the
J-band width, while for the second one it is of the order of the full J-band
width. We discuss also the experimental data on the fast low-temperature
exciton-exciton annihilation reported recently by I. G. Scheblykin {\em et al},
J. Phys. Chem. B {\bf 104}, 10949 (2000).
