Let G be a graph and suppose that for each vertex v of G, there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. M. Ghebleh and E. S. Mahmoodian characterized uniquely 3-List colorable complete multipartite graphs except for nine graphs. Recently, except for graph K 2,3,4, the other eight graphs were shown not to be uniquely 3-list colorable by W. He and Y. Shen, etc. In this paper, it is proved that K 2,3,4 is not uniquely 3-list colorable, and then the uniquely 3-list colorable complete multipartite graphs are characterized completely.