The center of M83, a barred starburst galaxy with a double nucleus, has been observed in the CO (J = 2-1) and CO (J = 3-2) lines with the Submillimeter Array. The molecular gas shows a distribution and kinematics typical for barred galaxies at ~1 kpc radii but reveals unusual kinematics around the double nucleus in the central ~300 pc. Our CO velocity data show that the visible nucleus in M83 is at least 3'' (65 pc) away from the galaxy's dynamical center, which most likely coincides with the center of symmetry previously determined in the K band and is suggested to host another nucleus. We discovered high-velocity molecular gas associated with the visible off-center nucleus and also found a steep velocity gradient across it. We attribute these features to a gas disk rotating around the off-center nucleus, which may be the remnant of a small galaxy accreted by M83. The dynamical mass of this component is estimated to be 3 × 10⁸ M within a radius of 40 pc. The dynamical perturbation from the off-center nucleus may have played a key role in shaping the lopsided starburst.