  Using deep Chandra ACIS observation data for Cygnus A, we report evidence of
non-thermal X-ray emission from radio lobes surrounded by a rich intra-cluster
medium (ICM). The diffuse X-ray emission, which are associated with the eastern
and western radio lobes, were observed in a 0.7--7 keV Chandra$ ACIS image. The
lobe spectra are reproduced with not only a single-temperature Mekal model,
such as that of the surrounding ICM component, but also an additional power-law
(PL) model. The X-ray flux densities of PL components for the eastern and
western lobes at 1 keV are derived as 77.7^{+28.9}_{-31.9} nJy and
52.4^{+42.9}_{-42.4} nJy, respectively, and the photon indices are
1.69^{+0.07}_{-0.13} and 1.84^{+2.90}_{-0.12}, respectively. The non-thermal
component is considered to be produced via the inverse Compton (IC) process, as
is often seen in the X-ray emission from radio lobes. From a re-analysis of
radio observation data, the multiwavelength spectra strongly suggest that the
seed photon source of the IC X-rays includes both cosmic microwave background
radiation and synchrotron radiation from the lobes. The derived parameters
indicate significant dominance of the electron energy density over the magnetic
field energy density in the Cygnus A lobes under the rich ICM environment.
