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For a graph $G$, we show that if $mad(G)<m$, then $\chi'_\ell(G)\leq\Delta+1$ where $m$ depends upon $\Delta$ and $\chi'_\ell(G)$ is thelist-chromatic index of $G$. When $\Delta\leq 20$ the value of $m$ is close to$\frac{1}{2}\Delta$, but as $\Delta$ increases $m$ becomes asymptotic to about$\frac{1}{4}\Delta+5$.

List-Edge-Coloring with Bounded Maximum Degree