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 × 106 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 the magnetic field from the galactic disk. Enhanced diffuse X-ray emission is apparently enclosed by numerous Hα-emitting loops blistered out from the central disk of the galaxy, as is evident in a comparison with our deep Hubble Space Telescope imaging.