Mixed-criticality systems have tasks with different criticality levels running on the same hardware platform. Today's DRAM controllers cannot adequately satisfy the often conflicting requirements of tightly bounded worst-case latency for critical tasks and high performance for non-critical real-time tasks. We propose a DRAM memory controller that meets these requirements by using bank-aware address mapping and DRAM command-level priority-based scheduling with preemption. Many standard DRAM controllers can be extended with our approach, incurring no performance penalty when critical tasks are not generating DRAM requests. Our approach is evaluated by replaying memory traces obtained from executing benchmarks on an ARM ISA-based processor with caches, which is simulated on the gem5 architecture simulator. We compare our approach against previous TDM-based approaches, showing that our proposed memory controller achieves dramatically higher performance for non-critical tasks, without any significant impact on the worstcase latency of critical tasks.