There exists a positive constant $C_{1}$ (independent of $\epsilon$) such that, for metrics as above with fibre volume $\epsilon$, the diameters of the fibres over $D^{\prime}$ are bounded above by $C_{1}(\epsilon\,\log{\epsilon^{-1}})^{1/2}$. Moreover, there exists a second constant $C_{2}$ such that the diameter $d(\epsilon)$ of the singular fibre is at least $C_{2}(\epsilon\,\log{\epsilon^{-1}})^{1/2}$.