Summary: Let \(G\) be a simple graph with vertex set \(V(G)\) and \(\alpha\) a real number other than 0 and 1. The first general Zagreb index of \(G\) is defined as \(M_1^\alpha(G)=\sum_{v\in V(G)}d(v)^\alpha\) where \(d(v)\) is the degree of \(v\). We determine the minimum value of the first general Zagreb index in the case \(\alpha\in(-\infty,0)\cup(1,+\infty)\), and the maximum value of the first general Zagreb index in the case \(\alpha\in(0,1)\), for bipartite graphs with a given number of vertices and edges.