  We give a complete description of the Poisson boundary of wreath products
$A\wr B= \bigoplus_{B} A\rtimes B$ of countable groups $A$ and $B$, for
probability measures $\mu$ with finite entropy where lamp configurations
stabilize almost surely. If, in addition, the projection of $\mu$ to $B$ is
Liouville, we prove that the Poisson boundary of $(A\wr B,\mu)$ is equal to the
space of limit lamp configurations, endowed with the corresponding hitting
measure. In particular, this answers an open question asked by Kaimanovich, and
Lyons-Peres, for $B=\mathbb{Z}^d$, $d\ge 3$, and measures $\mu$ with a finite
first moment.
