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 semimajor axis ≃11 (≃100 kpc). This structure apparently represents intracluster gas that has been swept up and compressed by a cavity inflated 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 ≃12 (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' (≃720 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' (≃1 Mpc) to the northwest of Cygnus A. The X-ray spectrum of the integrated intracluster gas imaged on the S3 chip (dimensions 8' × 8' = 720 × 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 × 1044 ergs s-1, respectively.