Summary: Let \((u_n)\) be a sequence of fuzzy numbers and \((p_n)\) be a sequence of nonnegative numbers such that \(p_0 > 0\) and \[ P_n := \sum^n_{k=0}p_k\to\infty\text{ as }n\to\infty. \] The weighted mean of \((u_n)\) is defined by \[ t_n :=\frac{1}{P_n}\sum^n_{k=0}p_ku_k\text{ for }n = 0, 1, 2,\dots\] It is well known that convergence of \((u_n)\) implies that of the sequence \((t_n)\) of its weighted means. However, the converse of this implication is not true in general. In this paper, we investigate under which conditions convergence of \((u_n)\) follows from its weighted mean summability. We prove a Tauberian theorem including condition of slow decrease with respect to the weighted mean summability method for sequences of fuzzy numbers.