This article presents a chance-constrained rollover-free manipulation planning method for robotic arms under payload mass uncertainty. The corresponding motion planning problem is stated as a chance-constrained nonlinear optimal control problem (NOCP) subject to kinematics and rollover stability constraints. The latter takes the form of a chance constraint that ensures a certain probability of the robot maintaining dynamic rollover stability in the presence of payload mass uncertainty. To achieve efficient solutions to the NOCP, a novel geometric bound for the stability region is derived. The novel bound is then utilized to modify the rollover-stability constraint. To showcase its benefit, comparisons between the proposed bound of the probabilistic rollover-stability measure and the naive noise model are provided through statistical analysis. The formulation's practicality is demonstrated through experiments with a Kinova Jaco2 arm mounted on a free-to-rollover platform. Results demonstrate greater robustness of the robot's motion plan to mass uncertainty and computational efficiency of the trajectory generation.