In a recent paper, Harary and Kabell [5] discuss the problem of determining the line index of balance of a signed graph. They establish a theorem which suggests a computational procedure solving this problem. The procedure is based upon a simple modification of their linear algorithm for detecting balance in signed graphs. Unfortunately, this procedure would involve considering all spanning trees of a given signed graph, the number of which may grow exponentially with the size of the graph. The question arises whether the problem of determining the line index of a signed graph can be solved efficiently {i.e., in polynomial time). We show that the problem is indeed difficult, being NP-complete in the general case.