We continue the investigation started in a previous paper, on
weak convergence to infinitely divisible distributions with finite
variance. In the present paper, we study this problem for some
weakly dependent random variables, including in particular
associated sequences. We obtain minimal conditions expressed in
terms of individual random variables. As in the i.i.d. case, we
describe the convergence to the Gaussian and the purely
non-Gaussian parts of the infinitely divisible limit. We also
discuss the rate of Poisson convergence and emphasize the special
case of Bernoulli random variables. The proofs are
mainly based on Lindeberg's method.