The authors prove the following rigidity theorem for homogeneous-rational manifolds of rank 1 under algebraic deformations: Let \(f:\quad X\to Y\) be a proper morphism of complex algebraic varieties, where Y is regular and all fibers of f are smooth algebraic varieties of constant dimension. Assume that for a closed point \(z\in Y\) the fiber \(V:=f^{-1}(z)\) is a homogeneous-rational manifold of rank 1. Then \(f^{-1}(y)\cong V\) for all closed points \(y\in Y\). Furthermore there is an integer \(r>0\) such that \({\mathcal O}_{f^{-1}(z)}(r)\) extends to an invertible sheaf H on X and \(X\hookrightarrow {\mathbb{P}}(f_*H)\).