We present an efficient method for motion control of redundant robots performing multiple prioritized tasks in the presence of hard bounds on joint range, velocity, and acceleration/ torque. This is an extension of our recently proposed SNS (Saturation in the Null Space) algorithm developed for single tasks. The method is defined at the level of acceleration commands and proceeds by successively discarding one at a time the commands that would exceed their bounds for a task of given priority, and reintroducing them at their saturated levels by projection in the null space of a suitable Jacobian associated to the already considered tasks. When processing all tasks in their priority order, a correct preemptive strategy is realized in this way, i.e., a task of higher priority uses in the best way the feasible robot capabilities it needs, while lower priority tasks are accommodated with the residual capability and do not interfere with the execution of higher priority tasks. The algorithm automatically integrates a multi-task least possible scaling strategy, when some ordered set of original tasks is found to be unfeasible. Simulation and experimental results on a 7-dof lightweight KUKA LWR IV robot illustrate the good performance of the method.