  Let $G$ be a simply connected, solvable Lie group and $\Gamma$ a lattice in
$G$. The deformation space $\mathcal{D}(\Gamma,G)$ is the orbit space
associated to the action of $\Aut(G)$ on the space $\mathcal{X}(\Gamma,G)$ of
all lattice embeddings of $\Gamma$ into $G$. Our main result generalises the
classical rigidity theorems of Mal'tsev and Sait\^o for lattices in nilpotent
Lie groups and in solvable Lie groups of real type. We prove that the
deformation space of every Zariski-dense lattice $\Gamma$ in $G$ is finite and
Hausdorff, provided that the maximal nilpotent normal subgroup of $G$ is
connected. This implies that every lattice in a solvable Lie group virtually
embeds as a Zariski-dense lattice with finite deformation space. We give
examples of solvable Lie groups $G$ which admit Zariski-dense lattices $\Gamma$
such that $\mathcal{D}(\Gamma,G)$ is countably infinite, and also examples
where the maximal nilpotent normal subgroup of $G$ is connected and
simultaneously $G$ has lattices with uncountable deformation space.
