We report the results of the initial Far Ultraviolet Spectroscopic Explorer observations of molecular hydrogen (H2) in translucent clouds. These clouds have greater optical depth than any of the diffuse clouds previously observed for far-UV H2 absorption and provide new insights into the physics and chemistry of such regions. Our initial results involve observations of HD 73882, a well-studied southern hemisphere star lying behind substantial interstellar material (EB-V = 0.72; AV = 2.44). We find a total H2 column density N(H2) = 1.2 × 1021 cm-2 about 3 times larger than the values for diffuse clouds previously measured in the far-UV. The gas kinetic temperature indicated by the ratio N(J = 1)/N(J = 0) is 58 ± 10 K. With the aid of ground-based data to calculate an appropriate multicomponent curve of growth, we have determined column densities for all rotational levels up to J = 7. The J ≥ 2 states can be reasonably fitted with a rotational excitation temperature of 307 ± 23 K. Both the kinetic and rotational temperatures are similar to those found in previous investigations of diffuse clouds. The ratios of carbonaceous molecules to hydrogen molecules are also similar to ratios in diffuse clouds, suggesting a similar chemistry for this line of sight.