We classify simple weight modules over the Drinfeld double \({\mathcal {D}}\) of the bosonization of the Jordan plane that have at least one finite-dimensional weight space and determine their annihilators. We show that every primitive ideal, except \({\mathfrak {m}}_{+}{\mathcal {D}}\) , appears as the annihilator of some such module. Furthermore, we construct a family of simple weight modules whose annihilators are \({\mathfrak {m}}_{+}{\mathcal {D}}\) , none of which have finite-dimensional weight spaces.
