The supernova remnant G347.3À0.5 (J1713.7À3946) is known as one of the unique SNRs that emit TeV -rays, as well as nonthermal X-rays. We present a detailed study of molecular gas toward this SNR obtained with the 4 m millimeter and submillimeter telescope NANTEN at an angular resolution of 2A6. This study has revealed that several intensity peaks and the overall distribution of the molecular gas with radial velocities from À12 to À3 km s À1 show a remarkably good correlation with the X-ray features, strongly supporting the recently derived kinematic distance around 1 kpc, as opposed to the 6 kpc previously claimed. In addition, we show that absorption of X-rays is caused by local molecular gas at softer X-ray bands. Subsequent measurements of the submillimeter J ¼ 3 2 transition of CO made with the ASTE 10 m and CSO 10.4 m submillimeter telescopes toward three of the molecular intensity peaks have revealed higher excitation conditions, most likely higher temperatures above $30 K , in contrast to the typical gas temperature, 10 K, in low-mass dark clouds. This temperature rise is most likely caused by enhanced heating by the high-energy events in the SNR, where possible mechanisms include heating by X-rays, -rays, and/or cosmic-ray protons, although we admit that additional radiative heating by young protostars embedded may be working as well. In one of the CO peaks, we have confirmed the presence of broad molecular wings of $20 km s À1 velocity extent in the CO J ¼ 3 2 transition. Two alternative interpretations for the wings are presented; one is shock acceleration by the blast wave, and the other is molecular outflow driven by an embedded protostar. The SNR evolution is well explained as the free expansion phase based on the distance of 1 kpc. The molecular data set should be valuable for making a further detailed comparison with the -ray and X-ray distributions in order to examine the cosmic-ray acceleration quantitatively. Subject headingg s: cosmic rays -ISM: individual (SNR J1713.37À3946) -ISM: kinematics and dynamicssupernova remnants