  We demonstrate the fabrication of ultra-low-loss, all-fiber Fabry-P\'erot
cavities containing a nanofiber section, optimized for cavity quantum
electrodynamics. By continuously monitoring the finesse and fiber radius during
fabrication of a nanofiber between two fiber Bragg gratings, we are able to
precisely evaluate taper transmission as a function of radius. The resulting
cavities have an internal round-trip loss of only 0.31% at a nanofiber waist
radius of 207 nm, with a total finesse of 1380, and a maximum expected internal
cooperativity of $\sim$ 1050 for a cesium atom on the nanofiber surface. Our
ability to fabricate such high-finesse nanofiber cavities may open the door for
the realization of high-fidelity scalable quantum networks.
