We study the weak amenability of a general measure algebra M(X) on a locally compact space X. First we show that not all general measure multiplications are separately weak* continuous; moreover, under certain conditions, weak amenability of M(X)** implies weak amenability of M(X). The main result of this paper states that there is a general measure algebra M(X) such that M(X) and M(X)** are weakly amenable without X being a discrete topological space.