Let \(G\) be a complex Lie group and \(M\) a compact connected Kähler manifold. Consider \(0={\mathcal E}_0 \subset {\mathcal E}_1 \subset \dots {\mathcal E}_{l-1} \subset {\mathcal E}_l=T\), the Harder-Narasimhan filtration of the holomorphic tangent bundle \(T\) of \(M\). The following result is proven: Theorem. If \(\text{deg}(T/{\mathcal E}_{l-1})\geq 0\), then a holomorphic principal \(G\)-bundle \(P\) on \(M\) admitting a compatible holomorphic connection is semi-stable. Moreover, if \(\text{deg}(T/{\mathcal E}_{l-1})>0\), then such a bundle \(P\) actually admits a compatible flat \(G\)-connection. In particular, if \(T\) is semi-stable with \(\text{deg}(T)>0\), then a \(G\)-bundle \(P\) on \(M\) with a holomorphic connection admits a compatible flat connection.