We present an analysis of the Chandra Advanced CCD Imaging Spectrometer (ACIS) observation of the intracluster gas associated with the cluster of galaxies surrounding Cygnus A. The dominant gaseous structure is a roughly elliptical (presumably prolate spheroidal in three dimensions) feature with semi-major axis similar or equal to 1'.1 (similar or equal to100 kpc). This structure apparently represents intracluster gas that has been swept up and compressed by a cavity innated in this gas by relativistic material that has passed through the ends of the radio jets. The X-ray-emitting gas shows this prolate spheroidal morphology to similar or equal to1'.2 (110 kpc) from the radio galaxy but is spherical on larger scales. The X-ray emission from the intracluster gas extends to at least 8' (similar or equal to720 kpc) from the radio galaxy, and a second, extended source of X-ray emission (probably associated with a second cluster of galaxies) is seen some 12' (similar or equal to1 Mpc) to the northwest of Cygnus A. The X-ray spectrum of the integrated intracluster gas imaged on the S3 chip (dimensions 8' x 8' = 720 x 720 kpc), excluding the contribution from the radio galaxy and other compact sources of X-ray emission, has a gas temperature, metallicity, and unabsorbed 2-10 keV rest-frame luminosity of 7.7 keV, 0.34 times solar, and 3.5 x 10(44) ergs s(-1), respectively. We have deprojected the X-ray spectra taken from 12 elliptical and circular annuli in order to derive a run of temperature, metallicity, density, and pressure as a function of radius. The temperature of the X-ray-emitting gas drops from similar or equal to8 keV more than 100 kpc from the center to similar or equal to5 keV some 80 kpc from the center, with the coolest gas immediately adjacent to the radio galaxy. "Belts" of slightly cooler (similar or equal to4 keV) X-ray-emitting gas run around the minor dimension of the cavity created by the radio source, while the limb-brightened edges of the cavity are slightly hotter (similar or equal to6 keV), perhaps as a result of heating by a bow shock driven by the probably expanding cavity into the intracluster gas. There is a metallicity gradient in the X-ray-emitting gas, with the highest metallicities (similar tosolar) found close to the center, decreasing to similar to0.3 times solar in the outer parts. We have used the assumption of hydrostatic equilibrium to derive a total cluster mass within 500 kpc of 2.0 x 10(14) M-circle dot and 2.8 x 10(14) M-circle dot for constant and centrally decreasing temperature profiles, respectively. The total mass of X-ray-emitting gas within the same radius is 1.1 x 10(13) M-circle dot. Thus, the gas fraction of the cluster within 500 kpc is 0.055 and 0.039 for the constant and centrally decreasing temperature profiles, respectively.