  Let $X$ be a generic curve of genus $g$ defined over an algebraically closed
field $k$ of characteristic $p\geq 0$. We show that for $n$ sufficiently large
there exists a tame rational map $f:X\to \PP^1_k$ with monodromy group $A_n$.
This generalizes a result of Magaard--V\"olklein to positive characteristic.
