The author shows that the leaf space \(M/{\mathcal F}\) of a 1-dimensional regular geodesic transversely Kähler foliation \({\mathcal F}\) of a \((2n+1)\)-dimensional \((n>2)\) compact simply-connected Riemannian manifold \(M\) of positive sectional curvature is biholomorphic to \(\mathbb{C} P^ n\) provided that the transverse conformal curvature tensor \(B\) of \(\mathcal F\) is parallel. A somewhat similar result for foliated Sasakian manifolds is obtained.