The {\em bandwidth problem} is the problem of numbering the vertices of a given graph $G$ such that the maximum difference between the numbers of adjacent vertices is {\em minimal}. The problem has a long and varied history and is known to be $NP$-hard Papadimitriou \cite{Pap76}. Recently for dense graphs a constant ratio approximation algorithm for this problem has been constructed in Karpinski, Wirtgen and Zelikovsky \cite{KaWiZe97}. In this paper we prove that the bandwidth problem on the dense instances remains $NP$-hard.