We demonstrate a fiber-integrated Fabry-Perot cavity formed by attaching a pair of dielectric metasurfaces to the ends of a hollow-core photonic-crystal fiber segment. The metasurfaces consist of perforated membranes designed as photonic-crystal slabs that act as planar mirrors but can potentially allow injection of gases through their holes into the hollow core of the fiber. We have so far observed cavities with finesse of ~11 and Q factors of ~$4.5 times 10^5$, but much higher values should be achievable with improved fabrication procedures. We expect this device to enable development of new fiber lasers, enhanced gas spectroscopy, and studies of fundamental light-matter interactions and nonlinear optics.