  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\times10^6 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.
