The J = 27-26 and J = 30-29 transitions of CO have been detected in the Orion molecular cloud. These detections, together with an improved measurement of the J = 21-20 transition, allow estimation of the temperature and density of the shocked material and the fractional abundance of CO. By solving the equations of detailed balance for J less than or equal to 50 and fitting the data to a two-component model consistent with earlier 2-micron and 12-micron H2 observations, it is shown that the hot (2000 K) component has a density of approximately 1.0 x 10 to the 6th/cu cm, while the cooler component lies in the range of 400-1000 K and is 2-5 times more dense. Approximately 25% of the carbon is in the form of CO. The spatial extent of the hot CO has also been examined by observing the J = 21-20 transition at a number of positions, and it is found to be distributed similarly to the H2 lines. In addition, a search for J = 21-20 and J = 22-21 CO emission from six other sources has resulted in 3-standard-deviation upper limits of a factor of 10 below the intensity of the Orion lines.