In a 52 ks-long Chandra ACIS-S observation of the nearby starburst galaxy NGC 3628 , obtained to study the starburst-driven outflow from this galaxy , we have detected a very luminous ( L _ { X } \approx 1.1 \times 10 ^ { 40 } \hbox { $ { \thinspace erg } { \thinspace s } ^ % { -1 } $ } in the 0.3 – 8.0 keV energy band ) point source located at least 20 \arcsec ( \sim 970 pc ) from the nucleus of the galaxy . No radio , optical or near-IR counterpart to this source has been found . This is most probably the reappearance of the strongly-variable X-ray-luminous source discovered by Dahlem , Heckman & Fabbiano ( 8 ) , which faded by a factor \gtrsim 27 between December 1991 and March 1994 ( at which point it had faded below the detection limit in a ROSAT HRI observation ) . This source is clearly a member of an enigmatic class of X-ray sources that are considerably more luminous than conventional X-ray binaries but less luminous than AGN , and which are not found at the dynamical center of the host galaxy . The Chandra spectrum is best-fit by an absorbed power law model with a photon index of \Gamma = 1.8 \pm { 0.2 } , similar to that seen in Galactic BH binary candidates in their hard state . Bremsstrahlung models or multi-color disk models ( the favored spectral model for objects in this class based on ASCA observations ) can provide statistically acceptable fits only if the data at energies E > 5 keV is ignored . This is one of the first X-ray spectra of such an object that is unambiguously that of the source alone , free from the spectral contamination by X-ray emission from the rest of the galaxy that affects previous spectral studies of these objects using ASCA .