Observations have been made of two quadrupole purely rotational transitions of molecular hydrogen, the 0-0 S(7) and S(5) lines, and of several previously unobserved vibration-rotation hydrogen lines in the Orion molecular cloud. The line ratios indicate a 2.1-micron extinction of about 2 mag. A comparison of the populations of three vibrational levels shows a distribution of excitation temperatures between 1000 and 3000 K. These values are consistent with the theory that hydrogen is thermally excited in shock-heated gas which cools after passage of the shock front. The total luminosity of the hot molecular hydrogen is 200 plus or minus 80 solar luminosity, and the mass of the emitting gas is of the order 0.05 solar mass. The mass and cooling times of the hot hydrogen indicate that at least 10 solar mass of the material has been heated over the lifetime of the outflow.