We report new CO observations and a detailed molecular-line study of the mixed morphology (MM) supernova remnant (SNR) G359.1-0.5, which contains six OH (1720 MHz) masers along the radio shell, indicative of shock-cloud interaction. Observations of 12CO and 13CO J:1-0 lines were performed in a 38x38 arcmin area with the on-the-fly technique using the Kit Peak 12 Meter telescope. The molecular study has revealed the existence of a few clumps with densities around 1,000 cm$^{-3}$ compatible in velocity and position with the OH (1720 MHz) masers. These clumps, in turn, appear to be part of a larger, elongated molecular structure ~34 arcmin long extending between -12.48 and +1.83 km/s, adjacent to the western edge of the radio shell. According to the densities and relative position with respect to the masers, we conclude that the CO clouds depict unshocked gas, as observed in other remnants with OH (1720 MHz) masers. In addition, we investigated the distribution of the molecular gas towards the adjacent gamma-ray source HESS J1745-303 but could not find any morphological correlation between the gamma-rays and the CO emission at any velocity in this region.
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