In a 14 ks ROSAT Position Sensitive Proportional Counter pointing we d
etected an extended (28 x 40 kpc) halo of hot tenuous gas around NGC 3
628, with a flat emission distribution and a temperature of T similar
or equal to 2 x 10(6) K. The soft X-ray luminosity of the halo gas in
the 0.1-2.0 keV band is similar to 8 x 10(39) ergs s(-1) and the total
soft X-ray luminosity of NGC 3628 is approximately 2.1 x 10(40) ergs
s(-1). The most likely heating source for the halo gas are the shocks
resulting from a supernova-driven outflow from the circumnuclear starb
urst propagating through the halo interstellar medium. Both the morpho
logy and the line ratios of associated optical emission-line gas are c
onsistent with this interpretation. In many respects the soft X-ray ha
lo of NGC 3628 resembles those of other nearby, edge-on starburst gala
xies like, e.g., M82 and NGC 253. Its size and radiative cooling time
indicate that halos like that of NGC 3628 have to be considered as pot
entially important contributors to the population of quasi-stellar obj
ect absorption-line systems. With two exceptions, most of the compact
X-ray sources seen in projection within the halo of NGC 3628 are proba
bly background active galactic nuclei. The remaining two sources might
be condensations in the shock-heated halo gas. The soft X-ray propert
ies of a few sources in the disk of NGC 3628 are consistent with those
of high-mass X-ray binaries or powerful young supernova remnants. The
PSPC spectrum of NGC 3627, which was within the field of view during
our observations, also exhibits the typical characteristics of a starb
urst galaxy. It can be approximated by a power-law + Raymond-Smith mod
el with a total soft X-ray luminosity of L(X) = 1.1 x 10(40) ergs s(-1
).