  We investigate to what extent a nilpotent Lie group is determined by its
$C^*$-algebra. We prove that, within the class of exponential Lie groups,
direct products of Heisenberg groups with abelian Lie groups are uniquely
determined even by their unitary dual, while nilpotent Lie groups of dimension
$\le 5$ are uniquely determined by the Morita equivalence class of their
$C^*$-algebras. We also find that this last property is shared by the filiform
Lie groups and the $6$-dimensional free two-step nilpotent Lie group.
