Embedding *-ordered domains into skew fields

We study the problem of embedding domains with *-orderings into skew fields. Assuming that the natural valuation associated to a *-ordered domain satisfies an Ore-type condition, we prove that the domain embeds in an order-preserving way into a *-ordered skew field. We call this the *-ordered version of the Dauns embedding theorem for domains with valuations. A number of concrete examples, where this result can be applied, is given. Moreover, a question of Murray Marshall regarding *-orderable groups is answered in the affirmative.