We present an analysis of the X-ray spectrum of the Galactic shell-type SNR G347.3 - 0.5 ( RX J1713.7 - 3946 ) . This SNR is a member of a growing class of SNRs which are dynamically young , shell-type sources that emit non-thermal X-rays from specific regions on their outer shells . By performing a joint spectral analysis of data from observations made of G347.3 - 0.5 using the \it { ROSAT } PSPC , the \it { ASCA } GIS and the \it { RXTE } PCA , we have fit the spectra of particular regions of this SNR ( including the bright northwestern and southwestern rims , the northeast rim and the interior diffuse emission ) over the approximate energy range of 0.5 through 30 keV . We find that fits to the spectra of this SNR over this energy range using the \it { SRCUT } model were superior to a simple power law model or the \it { SRESC } model . We find that the inclusion of a thermal model with the \it { SRCUT } model helps to improve the fit to the observed X-ray spectrum : this represents the first detection of thermal X-ray emission from G347.3 - 0.5 . Thermal emission appears to be more clearly associated with the diffuse emission in the interior of the SNR than with the bright X-ray emitting rims . A weak emission feature seen near 6.4 keV in the \it { RXTE } PCA spectrum most likely originates from diffuse X-ray emission from the surrounding Galactic Ridge rather than from G347.3 - 0.5 itself . We have analyzed our \it { RXTE } PCA data to search for pulsations from a recently discovered radio pulsar ( PSR J1713 - 3949 ) which may be associated with G347.3 - 0.5 , and we do not detect any X-ray pulsations at the measured radio period of 392 ms . Using the best-fit parameters obtained from the \it { 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 B = 10 \mu G ) , consistent with the results of ( ) . We present a broadband ( radio to \gamma -ray ) photon energy-flux spectrum for the northwestern rim of G347.3 - 0.5 , where we have fit the spectrum using a more sophisticated synchrotron-inverse Compton model with a variable magnetic field strength . Our fit derived from this model yields a maximum energy of only 8.8 _ { -3.4 } ^ { +4.1 } TeV for the accelerated cosmic-ray electrons and a much greater magnetic field strength of 150 _ { -80 } ^ { +250 } \mu G : however , our derived ratio of volumes for TeV emission and X-ray emission based on this fit – V _ { TeV } / V _ { X - ray } \approx 1000 – is too large to be physically acceptable . We argue that neither non-thermal bremsstrahlung nor neutral pion paritcle decay can adequately explain the TeV emission from this rim , and therefore the physical process responsible for this emission at this site 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 .