We consider the problem of reaching agreement in synchronous systems under a fault model whose severity lies between Byzantine and crash faults. For these "mortal" Byzantine faults, we assume that faulty processes take a finite number of arbitrary steps before they eventually crash. After discussing several application examples where this model is justified, we present and prove correct a consensus algorithm that tolerates a minority of faulty processes; i.e., more faults can be tolerated compared to classic Byzantine faults. We also show that the algorithm is optimal regarding the required number of processes and that no algorithm can solve consensus with just a majority of correct processes in a bounded number of rounds under our fault assumption. Finally, we consider more restricted fault models that allow to further reduce the required number of processes.