Cosserat rods have become an increasingly popular framework for simulating complex bending and twisting in thin elastic rods, used for hair, tree, and yarn-level cloth models. However, traditional approaches often encounter significant challenges in robustly and efficiently solving for valid quaternion orientations, even when employing small time steps or computationally expensive global solvers. We introduce stable Cosserat rods, a new solver that can achieve high accuracy with high stiffness levels and maintain stability under large time steps. It is also inherently suitable for parallelization. Our key contribution is a split position and rotation optimization scheme with a closed-form Gauss-Seidel quasi-static orientation update. This solver significantly improves the numerical stability with Cosserat rods, allowing faster computation and larger time steps. We validate our method across a wide range of applications, including simulations of hair, trees, yarn-level cloth, slingshots, and bridges, demonstrating its ability to handle diverse material behaviors and complex geometries. Furthermore, we show that our method is orders of magnitude faster and more stable than alternative rod solvers, such as extended position-based dynamics and discrete elastic rods.