G328.4+0.2 is a moderately small (5 arcmin x 5arcmin) Galactic radio supernova remnant (SNR) at a distance of at least 17 kpc that has been long suggested to be Crab-like. Here we report on the detection with ASCA of the X-ray emission from the SNR. The X-ray source is faint with an observed flux of (6.0 +/- 0.8)E-13 erg/s/cm^2 over the 2-10 keV band. The emission is heavily cut-off at low energies and no flux is detected below 2 keV. Spectral analysis confirms that the column density to the source is indeed large, N_H ~ 1E23 atoms/cm^2, and consistent with the total column density of hydrogen through the Galaxy at this position. Good fits to the spectrum can be obtained for either thermal plasma or nonthermal power-law models, although the lack of detected line emission as well as other evidence argues against the former interpretation. The power-law index we find, alpha_P = 2.9 (+0.9,-0.8), is consistent with other Crab-like SNRs. In the radio band G328.4+0.2 is nearly as luminous as the Crab Nebula, yet in the X-ray band luminosity it is some 70 times fainter. Nevertheless its inferred soft X-ray band luminosity is greater than all but the brightest pulsar-powered synchrotron nebulae and implies that G328.4+0.2 contains a rapidly spinning, as yet undetected, pulsar that is losing energy at a rate of approximately 1E38 erg/s.
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