We present a Chandra X-ray observation that shows , unambiguously for the first time , the presence of a giant diffuse X-ray-emitting corona around the edge-on disk galaxy NGC 4631 . This corona , with a characteristic thermal temperature of 2 – 7 \times 10 ^ { 6 } K , extends as far as 8 kpc away from the galactic plane . The X-ray morphology resembles the radio halo of the galaxy , indicating a close connection between outflows of hot gas , cosmic rays , and magnetic field from the galactic disk . Enhanced diffuse X-ray emission is apparently enclosed by numerous H \alpha -emitting loops blistered out from the central disk of the galaxy , as is evident in a comparison with our deep Hubble Space Telescope imaging .