Recent ASCA observations of G347.3-0.5, an SNR discovered in the ROSAT All-Sky Survey, reveal nonthermal emission from a region along the northwestern shell (Koyama et al. 1997). Here we report on new pointed ASCA observations of G347.3-.5 which confirm this result for all the bright shell regions and also reveal similar emission, although with slightly different spectral properties, from the remainder of the SNR. Curiously, no thermal X-ray emission is detected anywhere in the remnant. We derive limits on the amount of thermal emitting material present in G347.3-0.5 and present new radio continuum, CO and infrared results which indicate that the remnant is distant and of moderate age. We show that our observations are broadly consistent with a scenario that has most of the supernova remnant shock wave still within the stellar wind bubble of its progenitor star, while part of it appears to be interacting with denser material. A point source at the center of the remnant has spectral properties similar to those expected for a neutron star and may represent the compact relic of the supernova progenitor.