Many safety-critical embedded systems are subject to certification requirements. However, only a subset of the functionality of the system may be safety-critical and hence subject to certification, the rest of the functionality is non safety-critical and does not need to be certified, or is certified to a lower level. The resulting mixed criticality system offers challenges both for static schedulability analysis and run-time monitoring. This paper considers a novel implementation scheme for fixed priority uniprocessor scheduling of mixed criticality systems. The scheme requires that jobs have their execution times monitored (as is usually the case in high integrity systems). An optimal priority assignment scheme is derived and sufficient response-time analysis is provided. The new scheme formally dominates those previously published. Evaluations illustrate the benefits of the scheme.