A maneuver to capture and detumble an orbiting space object using a chaser spacecraft equipped with a robotic manipulator is presented. In the proposed maneuver, the capture and detumble objectives are integrated into a unified set of terminal constraints. Terminal constraints on the end-effector's position and velocity ensure a successful capture, and a terminal constraint on the chaser's momenta ensures a post-capture chaser-target system with zero angular momentum. The manipulator motion required to achieve a smooth, impact-free grasp is gradually stopped after capture, equalizing the momenta across all bodies, rigidly connecting the two vehicles, and completing the detumble of the newly formed chaser-target system without further actuation. To guide this maneuver, an optimization-based approach that enforces the capture and detumble terminal constraints, avoids collisions, and satisfies actuation limits is used. The solution to the guidance problem is obtained by solving a collection of convex programming problems, making the proposed guidance approach suitable for onboard implementation and real-time use. This simultaneous capture and detumble maneuver is evaluated through numerical simulations and hardware-in-the-loop experiments. Videos of the numerically simulated and experimentally demonstrated maneuvers are included as Supplementary Material.