In this paper we address the problem of describing in explicit algebraic terms the collective structure of the coadjoint orbits of a connected, simply connected exponential solvable Lie group G. We construct a partition ℘ of the dual g* of the Lie algebra g of G into finitely many $\operatorname{Ad}^\ast(G)$-invariant algebraic sets with the following properties. For each Ω ∈ ℘, there is a subset Σ of Ω which is a cross-section for the coadjoint orbits in Ω and such that the natural mapping $\Omega/\operatorname{Ad}^\ast (G) \rightarrow \Sigma$ is bicontinuous. Each Σ is the image of an analytic $\operatorname{Ad}^\ast (G)$-invariant function P on Ω and is an algebraic subset of g*. The partition ℘ has a total ordering such that for each $\Omega \in \wp, \bigcup\{\Omega': \Omega' \leq \Omega\}$ is Zariski open. For each Ω there is a cone $W \subset \mathfrak{g}^\ast$, such that Ω is naturally a fiber bundle over Σ with fiber W and projection P. There is a covering of Σ by finitely many Zariski open subsets O such that in each O, there is an explicit local trivialization Θ: P-1(O) → W × O. Finally, we show that if Ω is the minimal element of ℘ (containing the generic orbits), then its cross-section Σ is a differentiable submanifold of g*. It follows that there is a dense open subset U of $G^\wedge$ such that U has the structure of a differentiable manifold and $G^\wedge \sim U$ has Plancherel measure zero.