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We present a general deterministic linear space reduction from priority queues to sorting implying that if we can sort up to n keys in S(n) time per key, then there is a priority queue supporting delete and insert in S(n)+O(1) time and find-min in constant time. Conversely, a priority queue can trivially be used for sorting: first insert all keys to be sorted, then extract them in sorted order by repeatedly deleting the minimum. Hence, asymptotically this settles the complexity of priority queues in terms of that of sorting. Besides nailing down the complexity of priority queues to that of sorting, and vice versa, we translate known sorting results into new results on priority queues for integers and strings in different computational models.