In this paper, we propose a time-efficient and exact algorithm for the problem of discovering the densest subgraph in big data. Current algorithms for solving this problem have three problems: i) they cannot handle the dilemma between the efficiency of handing big data and the precision of the discovered densest subgraph; ii) they cannot take advantage of both the parallel computing on MapReduce and in-memory computing on one computer; iii) their applicability to different kinds of graphs has not been discussed. Our proposed algorithm combines the MapReduce parallel computing with in-memory computing on one computer together to improve the efficiency and precision of discovering the densest subgraphs. The algorithm consists of two computational phases: i) the graph reduction in the MapReduce framework; ii) the densest subgraph discovery in memory. Further, we theoretically analyze the correctness of this algorithm and its applicability in different natural graphs. We conduct extensive experimental evaluations in a MapReduce framework on both massive real-world graphs and simulated graphs to test our algorithm in comparison with other algorithms. Experimental results show that our algorithm is more time-efficient and precise than other algorithms.