  We present the design and construction of a new experimental apparatus for
the trapping of single Ba$^+$ ions in the center of curvature of an
optical-quality hemispherical mirror. We describe the layout, fabrication and
integration of the full setup, consisting of a high-optical access monolithic
`3D-printed' Paul trap, the hemispherical mirror, a diffraction-limited
in-vacuum lens (NA = 0.7) for collection of atomic fluorescence and a
state-of-the art ultra-high vacuum vessel. This new apparatus enables the study
of quantum electrodynamics effects such as strong inhibition and enhancement of
spontaneous emission, and achieves a collection efficiency of the emitted light
in a single optical mode of 31%.
