  A coupled microwave-cavity system of cylindrical TM$_{010}$ single-mode has
been developed to search for dark matter axions around 10 $\mu {\rm eV}$(2.4
GHz) with the Rydberg-atom cavity detector at 10 mK range temperature. One
component of the coupled cavity (conversion cavity) made of oxygen-free
high-conductivity copper is used to convert an axion into a single photon with
the Primakoff process in the strong magnetic field, while the other component
(detection cavity) made of Nb is utilized to detect the converted photons with
Rydberg atoms passed through it without magnetic field.
  Top of the detection cavity is attached to the bottom flange of the mixing
chamber of a dilution refrigerator, thus the whole cavity is cooled down to 10
mK range to reduce the background thermal blackbody-photons in the cavity.
  The cavity resonant frequency is tunable over $\sim$ 15% by moving dielectric
rods inserted independently into each part of the cavities along the
cylindrical axis. In order to reduce the heat load from the higher temperature
side to the most cooled dilution refrigerator part, the tuning rod at the
conversion cavity is especially driven via the Kevlar strings with a stepping
motor outside the cryostat at room temperature.
  The lowest temperature achieved up to now is 12 $\sim$ 15 mK for the long
period operation and the loaded Q value at low temperature is 3.5 $\sim$ 4.5
$\times$ $10^4$ for the whole range of frequency tuning. Characteristics and
the performance of the coupled-cavity system are presented and discussed with
possible future improvements.
