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The Graph k-Cut problem is that of finding a set of edges of minimum total weight, in an edge-weighted graph, such that their removal from the graph results in a graph having at least k connected components. An algorithm with a running time of O(n^k2) for this problem has been known since 1988, due to Goldschmidt and Hochbaum. We show that the problem is hard for the parameterized complexity class W[1]. We also investigate the complexity of a related problem, Cutting A Few Vertices from a Graph, that asks for the minimum cost of separating at least k vertices from an edge-weighted connected graph. We show that this problem also is hard for W[1].