  The effect of a slow noise in non-diagonal matrix element, J(t), that
describes the diabatic level coupling, on the probability of the Landau-Zener
transition is studied. For slow noise, the correlation time, \tau_c, of J(t) is
much longer than the characteristic time of the transition. Existing theory for
this case suggests that the average transition probability is the result of
averaging of the conventional Landau-Zener probability, calculated for a given
constant J, over the distribution of J. We calculate a finite-\tau_c correction
to this classical result. Our main finding is that this correction is dominated
by sparse realizations of noise for which J(t) passes through zero within a
narrow time interval near the level crossing. Two models of noise, random
telegraph noise and gaussian noise, are considered. Naturally, in both models
the average probability of transition decreases upon decreasing \tau_c. For
gaussian noise we identify two domains of this fall-off with specific
dependencies of average transition probability on \tau_c.
