Combining efficient and safe control for safety-critical systems is challenging. Robust methods may be overly conservative, whereas probabilistic controllers require a trade-off between efficiency and safety. In this work, we propose a safety algorithm that is compatible with any stochastic Model Predictive Control method for linear systems with additive uncertainty and polytopic constraints. This safety algorithm uses the control inputs of a stochastic Model Predictive Control as long as a safe backup planner can ensure safety with respect to satisfying hard constraints subject to bounded uncertainty. Besides ensuring safe behavior, the proposed stochastic Model Predictive Control algorithm guarantees recursive feasibility and input-to-state stability of the system origin. The benefits of the safe stochastic Model Predictive Control algorithm are demonstrated in a numerical simulation, highlighting the advantages compared to purely robust or stochastic predictive controllers.