We report a discovery of shocked gas from the supernova remnant (SNR) G357.7+0.3. Our millimeter and submillimeter observations reveal broad molecular lines of CO(2-1), CO(3-2), CO(4-3), (CO)-C-13 (2-1), and (CO)-C-13 (3-2), HCO+, and HCN using the Heinrich Hertz Submillimeter Telescope, the Arizona 12 m Telescope, APEX, and the MOPRA Telescope. The widths of the broad lines are 15-30 km s(-1), and the detection of such broad lines is unambiguous, dynamic evidence showing that the SNR G357.7+0.3 is interacting with molecular clouds. The broad lines appear in extended regions (>4'.5 x 5'). We also present the detection of shocked H-2 emission in the mid-infrared but lacking ionic lines using Spitzer/IRS observations to map a few-arcminute area. The H2 excitation diagram shows a best fit with a two-temperature local thermal equilibrium model with the temperatures of similar to 200 and 660 K. We observed [C II] at 158 mu m and high-J CO(11-10) with the German Receiver for Astronomy at Terahertz Frequencies (GREAT) on the Stratospheric Observatory for Infrared Astronomy. The GREAT spectrum of [C II], a 3 sigma detection, shows a broad line profile with a width of 15.7 km(-1) that is similar to those of broad CO molecular lines. The line width of [C II] implies that ionic lines can come from a low-velocity C-shock. Comparison of H2 emission with shock models shows that a combination of two C-shock models is favored over a combination of C- and J-shocks or a single shock. We estimate the CO density, column density, and temperature using a RADEX model. The best-fit model with n(H-2) = 1.7 x 10(4) cm(-3), N(CO) = 5.6 x 10(16) cm(-2), and T = 75 K can reproduce the observed millimeter CO brightnesses.