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Computing an accurate mean of a set of time series is a critical task in applications like nearest-neighbor classification and clustering of time series. While there are many distance functions for time series, the most popular distance function used for the computation of time series means is the non-metric dynamic time warping (DTW) distance. A recent algorithm for the exact computation of a DTW-Mean has a running time of $\mathcal{O}(n^{2k+1}2^kk)$, where $k$ denotes the number of time series and $n$ their maximum length. In this paper, we study the mean problem for the move-split-merge (MSM) metric that not only offers high practical accuracy for time series classification but also carries of the advantages of the metric properties that enable further diverse applications. The main contribution of this paper is an exact and efficient algorithm for the MSM-Mean problem of time series. The running time of our algorithm is $\mathcal{O}(n^{k+3}2^k k^3 )$, and thus better than the previous DTW-based algorithm. The results of an experimental comparison confirm the running time superiority of our algorithm in comparison to the DTW-Mean competitor. Moreover, we introduce a heuristic to improve the running time significantly without sacrificing much accuracy.