A 14-arcsec FWHM beam has been used to observe the CO (3 to 21) emission in the central region of M82; the map thus obtained clearly shows the double-peaked structure, seen at other wavelengths, which has been interpreted as a circumnuclear ring. The first (C-13)O (3 to 2) detections in M82 are also reported. Using a large velocity gradient model, the six lines constituting the three lowest transitions of (C-12)O and (C-13)O allow a direct determination to be made of the properties of the molecular gas in M82; the ratios of these lines are rather unusual, and cannot be reproduced with a standard one-component model. A one-component model with a lowered (C-12)O/(C-13)O abundance ratio with values below 25, similar to those at the center of the Galaxy, yields a significantly better fit to all of the CO lines. 24 refs.