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