  We report on the fabrication and characterization of a Fabry-Perot
microcavity enclosing a thin diamond membrane at cryogenic temperatures. The
cavity is designed to enhance resonant emission of single nitrogen-vacancy
centers by allowing spectral and spatial tuning while preserving the optical
properties observed in bulk diamond. We demonstrate cavity finesse at cryogenic
temperatures within the range of F = 4,000-12,000 and find a sub-nanometer
cavity stability. Modeling shows that coupling nitrogen-vacancy centers to
these cavities could lead to an increase of remote entanglement success rates
by three orders of magnitude.
