  We describe the construction of an apparatus designed to realize a hybrid
quantum system comprised of a cryogenically cooled mechanical oscillator and
ultra-cold $^{87}$Rb atoms coupled via light. The outstanding feature of our
instrument is an in-situ adjustable asymmetric all-fiber membrane-in-the-middle
cavity located inside an ultra-high vacuum dilution refrigerator based
cryostat. We show that Bose-Einstein condensates of $N=2\times10^6$ atoms can
be produced in less than 20 s and demonstrate a single photon optomechanical
coupling strength of $g_0=2\pi\times9$ kHz employing a high-stress Si$_3$N$_4$
membrane with a mechanical quality factor $Q_{\rm m}>10^7$ at a cavity set-up
temperature of $T_{\rm MiM}=480$ mK.
