We have used the Owens Valley Millimeter Array to map (CO)-C-12 (J = 1-0) along a 3.5 kpc segment of the eastern spiral arm of M83 at resolutions of 6 ".5 x 3 ".5, 10 ", and 16 ". The CO emission in most of this segment lies along the sharp dust lane demarcating the inner edge of the spiral arm, but beyond a certain point along the arm the emission shifts downstream from the dust lane to become better aligned with the young stars seen in blue and H beta images. This morphology resembles that of the western arm of M100. Three possibilities, none of which is wholly satisfactory, are considered to explain the deviation of the CO arm from the dust lane: heating of the CO by UV radiation from young stars, heating by low-energy cosmic rays, and a molecular medium consisting of two (diffuse and dense) components that react differently to the density wave. Regardless, the question of what CO emission traces along this spiral arm is a complicated one. Masses based on CO emission and the virial theorem for 10 emission features roughly agree and are in the range 1.5-16 x 10(6) M.. These are lower than the masses of giant molecular associations in M51, but the discrepancy is probably due to the much higher linear resolution of these observations. Despite the uncertainty in what CO emission is tracing, we do not require a conversion factor of CO brightness to H-2 column density much different from the standard Galactic value if these structures are bound. Surprisingly, for the two fields where we can compare with single-dish data, only 2%-5% of the single-dish flux is seen in our observations. A possible explanation is that M83 contains much smoothly distributed molecular gas that is resolved out by the interferometer. Strong tangential streaming is observed where the arm crosses the kinematic major axis of the galaxy, implying that the shear becomes locally prograde in the arms. The amplitude of the tangential streaming is used along with a low-resolution single-dish radial profile of CO emission to infer a very high gas surface density of about 230 M. pc(-2) and an arm-interarm contrast greater than 2.3 in the part of the arm near the major axis. Using two different criteria, we find that the gas at this location is well above the threshold for gravitational instability-much more clearly so than in either M51 or M100. This finding is consistent with the unusually high H alpha surface brightness and star formation efficiency in M83: star formation may be particularly active because of strong gravitational instabilities