A Paley-Wiener theorem for the inverse spherical transform is proved for noncompact semisimple Lie groups $G$ which are either of rank one or with a complex structure. Let $K$ be a fixed maximal compact subgroup of $G$. For each $K$-bi-invariant function $f$ in the Schwartz space on $G$, consider the function $\tilde f$ defined on a fixed Weyl chamber $\mathfrak a^+$ by $\tilde f(H):= \Delta(H) \,f( \exp H)$. Here $\Delta(H) := \prod_{\alpha \in \Sigma^+} \left( \sinh \alpha(H) \right)^{m_{\alpha}/ 2}$, \linebreak where $\Sigma^+$ is the set of positive restricted roots and $m_{\alpha}$ is the multiplicity of the root $\alpha$. The $K$-bi-invariant functions $f$ whose spherical transform has compact support are identified as those for which $\tilde f$ extends holomorphically and with a specific growth to a certain subset of the complexification $\mathfrak a_c$ of $\mathfrak a$. The proof of the theorem in the rank-one case relies on the explicit inversion formula for the Abel transform.