Let $\pi:X\to B$ be an elliptic fibration with periods $\{1,\tau(y)\}$, over the open disc $B$ of radius $R$ in ${\bf C}$, with ${\rm Im}\,\tau(y)>0$, and let $B_{0}\subset B$ denote a smaller disc of radius $R_{0}<R$. Suppose we have a sequence on $X$ of $S^{1}$-invariant Ricci-flat metrics $g_{i}$ in canonical form (and with constant volume form), for which the volume $\epsilon_{i}:=\epsilon(g_{i})$ of the fibres tends to zero as $i\to\infty$. Then on $\pi^{-1}(B_{0})$ we have $W_{i}:=W(g_{i})\to 0$ uniformly as $i\to\infty$. On a fixed fibre, with periods $\{1,\tau\}$, we have the stronger statement that $\epsilon_{i}^{-1}W_{i}\ {\rm Im}\,\tau\to 1$ uniformly as $i\to\infty$.