Dixmier always solves this as follows, e.g. in C*-algebras — surely one possible example of good exposition (E. C. Lance’s translator‘s preface: “With is clear and straightforward style, this remains the best book from which to learn about C*-algebras”):
16.1. The compact group associated with a topological group
16.1.1. Theorem. Let $G$ be a topological group. There exists a compact group $\Sigma$ and a continuous morphism $\alpha:G\to\Sigma$ possessing the following property: (...). Furthermore, the pair $(\Sigma,\alpha)$ is determined up to isomorphism by this property.
(... Long proof goes here ...)
16.1.2. Definition. The group $\Sigma$ is called the compact group associated with $G$, and $\alpha$ is called the canonical morphism of $G$ into $\Sigma$.
The book has about two dozen “X.Y.2” definitions like this — e.g. no less than six over pp. 116–123 (disjoint representations, factor representation, quasi-equivalent representations, type I, multiplicity-free, multiplicity), and the almost last statement in the book is a definition (Plancherel measure).
Added: I took the above example as perhaps the closest to yours (constructed object). In fact Dixmier’s relative Bourbaki does yours exactly, just the same down to the use of a subtitle (Algebra I.2.4 “Monoid of fractions”, I.8.12 “Rings of fractions”, I.9.4 “The field of rational numbers”):
12. Rings of fractions
Theorem 4. (...)
Definition 8. The ring defined in Theorem 4 is called the ring of fractions (...)
Of course, this “classic” way is not the only one: I also agree with Nik Weaver’s answer, and with this contrasting epigraph in Reed & Simon, Functional Analysis (my emphasis): “A good definition should be the hypothesis of a theorem. (J. Glimm)”