This paper presents a finite position synthesis (f.p.s.) procedure of a spatial single-degree-of-freedom linkage that we call origami-evolved, spherically constrained spatial revolute–revolute (RR) chain here. This terminology is chosen because the linkage may be found from the mechanism equivalent of an origami folding pattern, namely, known as the Miura-ori folding. As shown in an earlier work, the linkage under consideration has naturally given slim shape and essentially represents two specifically coupled spherical four-bar linkages, whose links may be identified with spherical and spatial RR chains. This provides a way to apply the well-developed f.p.s. theory of these linkage building blocks in order to design the origami-evolved linkage for a specific task. The result is a spherically constrained spatial RR chain, whose end effector may reach three finitely separated task positions. Due to an underspecified spherical design problem, the procedure provides several free design parameters. These can be varied in order to match further given requirements of the task. This is shown in a design example with particularly challenging space requirements, which can be fulfilled due to the naturally given slim shape.