Optical antennas have been extensively employed to manipulate the photophysical properties of single-photon emitters. Coupling between an emitter and a given resonant mode of an optical antenna depends mainly on three parameters: spectral overlap, relative distance, and relative orientation between the emitter's transition dipole moment and the antenna. While the first two have already been extensively demonstrated, achieving full coupling control remains unexplored due to the challenges in manipulating at the same time both the position and orientation of single molecules. Here, we use the DNA origami technique to assemble a dimer optical antenna and position a single fluorescent molecule at the antenna gap with controlled orientation, predominately parallel or perpendicular to the antenna's main axis. We study the coupling for both conditions through fluorescence measurements correlated with scanning electron microscopy images, revealing a 5-fold higher average fluorescence intensity when the emitter is aligned with the antenna's main axis and a maximum fluorescence enhancement of ∼1400-fold. A comparison to realistic numerical simulations suggests that the observed distribution of fluorescence enhancement arises from small variations in the emitter orientation and gap size. This work establishes DNA origami as a versatile platform to fully control the coupling between emitters and optical antennas, trailblazing the way for self-assembled nanophotonic devices with optimized and more homogeneous performance.