We modify a construction of A. Lemin and V. Lemin to construct an ultrametric space $LW_\tau^\prime$ which is universal (in the sense of isometry) for ultrametric spaces of weight at most $\tau$. Under the singular cardinal hypothesis, a set-theoretic assumption whose negation is related to large cardinals, the weight of $LW_\tau^\prime$ is $\tau$ for all $\tau > \frak{c}$. This provided a solution to a problem raised by the Lemins.