  We establish that all rings of $S$-integers are universally definable in
function fields in one variable over certain ground fields including global and
non-archimedean local fields. That is, we show that the complement of such a
ring of $S$-integers is always a diophantine set. As a technical tool, we use a
reciprocity exact sequence for quadratic Witt groups in function fields over
almost arbitrary base fields (of any characteristic), which is new and of
potentially independent interest.
