  We prove in arbitrary characteristic that an immediate valued algebraic
function field $F$ of transcendence degree 1 over a tame field $K$ is contained
in the henselization of $K(x)$ for a suitably chosen $x\in F$. This eliminates
ramification in such valued function fields. We give generalizations of this
result, relaxing the assumption on $K$. Our theorems have important
applications to local uniformization and to the model theory of valued fields
in positive and mixed characteristic.
