(Abridged) We present an analysis of the X-ray spectrum of the Galactic shell-type supernova remnant (SNR) G347.3-0.5 (RX J1713.7-3946). By performing a joint spectral analysis of data from observations made of G347.3-0.5 using the ROSAT PSPC, the ASCA GIS and the RXTE PCA, we have fit the spectra of particular regions of this SNR (including the bright northwestern and southwestern rims, the northeastern rim and the interior diffuse emission) over the approximate energy range of 0.5 through 30 keV. Based on the parameters of the best fit to the spectra using the SRCUT model, we estimate the maximum energy of cosmic-ray electrons accelerated by the rims of G347.3-0.5 to be 19-25 TeV, assuming a magnetic field strength of 10 microGauss. We present a broadband (radio to gamma-ray) photon energy flux-spectrum for the northwestern rim of the SNR, using a synchrotron-inverse Compton model with a variable magnetic field strength to fit the spectrum. Our fit derived from this model yields a maximum energy of only 8.8 TeV for the accelerated cosmic-ray electrons and a magnetic field strength of 150 microGauss. However, our derived ratio of volumes for TeV emission and X-ray emission (approximately 1000) is too large to be physically acceptable. We argue that neither non-thermal bremsstrahlung nor neutral pion decay can adequately describe the TeV emission from this rim, and therefore the physical process responsible for this emission is currently uncertain. Finally, we compare the gross properties of G347.3-0.5 with other SNRs known to possess X-ray spectra dominated by non-thermal emission.
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