  We prove that each $3$-dimensional connected topological loop $L$ having a
solvable Lie group of dimension $\le 5$ as the multiplication group of $L$ is
centrally nilpotent of class $2$. Moreover, we classify the solvable
non-nilpotent Lie groups $G$ which are multiplication groups for
$3$-dimensional simply connected topological loops $L$ and $\hbox{dim} \ G \le
5$. These groups are direct products of proper connected Lie groups and have
dimension $5$. We find also the inner mapping groups of $L$.
