Rational curves on elliptic surfaces

We prove that a very general elliptic surface $\mathcal{E}\to {\mathbb{P}}^1$ over the complex numbers with a section and with geometric genus $p_g\ge 2$ contains no rational curves other than the section and components of singular fibers. Equivalently, if $E\mathrm{/}\mathbb{C}(t)$ is a very general elliptic curve of height $d\ge 3$ and if $L$ is a finite extension of $\mathbb{C}(t)$ with $L\cong \mathbb{C}(u)$, then the Mordell-Weil group $E(L)=0$.