The >100 GeV gamma-ray source, HESS J1713-381, apparently associated with the shell-type supernova remnant (SNR) CTB 37B, was discovered using H.E.S.S. in 2006. X-ray follow-up observations with Chandra were performed in 2007 with the aim of identifying a synchrotron counterpart to the TeV source and/or thermal emission from the SNR shell. These new Chandra data, together with additional TeV data, allow us to investigate the nature of this object in much greater detail than was previously possible. The new X-ray data reveal thermal emission from a ~4' region in close proximity to the radio shell of CTB 37B. The temperature of this emission implies an age for the remnant of ~5000 years (assuming a spherical Sedov expansion), disfavouring a suggested association with the supernova of AD 373. A bright (approx 7 x10^-13erg cm^-2 s^-1) and unresolved (<1'') source (CXOU J171405.7-381031) with a soft (Gamma ~ 3.3) non -thermal spectrum is also detected in coincidence with the radio shell. Absorption indicates a column density consistent with the thermal emission from the shell suggesting a genuine association rather than a chance alignment. The observed TeV morphology is consistent with an origin in the complete shell of CTB 37B. The relatively high apparent gas density of the CTB 37B environment (n < 1 cm^-3) and the lack of diffuse non-thermal X-ray emission, are suggestive of an origin of the gamma-ray emission via the decay of neutral pions produced in interactions of protons and nuclei rather than inverse Compton (IC) emission from relativistic electrons.< Réduire