We present a Chandra X-ray observation that shows, unambiguously for the fi
rst 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 te
mperature of (2-7) x 10(6) K, extends as far as 8 kpc away from the galacti
c plane. The X-ray morphology resembles the radio halo of the galaxy, indic
ating a close connection between outflows of hot gas, cosmic rays, and the
magnetic field from the galactic disk. Enhanced diffuse X-ray emission is a
pparently enclosed by numerous H alpha -emitting loops blistered out from t
he central disk of the galaxy, as is evident in a comparison with our deep
Hubble Space Telescope imaging.