We report the results of a search for molecular oxygen (O_2) toward the Orion Bar, a prominent photodissociation region at the southern edge of the H II region created by the luminous Trapezium stars. We observed the spectral region around the frequency of the O_2 NJ = 33-12 transition at 487 GHz and the 5_(4)-3_(4) transition at 774 GHz using the Heterodyne Instrument for the Far-Infrared on the Herschel Space Observatory. Neither line was detected, but the 3σ upper limits established here translate to a total line-of-sight O2 column density <1.5 × 10^(16) cm^(–2) for an emitting region whose temperature is between 30 K and 250 K, or <1 × 10^(16) cm^(–2) if the O_2 emitting region is primarily at a temperature of ≲100 K. Because the Orion Bar is oriented nearly edge-on relative to our line of sight, the observed column density is enhanced by a factor estimated to be between 4 and 20 relative to the face-on value. Our upper limits imply that the face-on O_2 column density is less than 4 × 10^(15) cm^(–2), a value that is below, and possibly well below, model predictions for gas with a density of 10^(4)-10^(5) cm^(–3) exposed to a far-ultraviolet flux 10^4 times the local value, conditions inferred from previous observations of the Orion Bar. The discrepancy might be resolved if (1) the adsorption energy of O atoms to ice is greater than 800 K; (2) the total face-on A V of the Bar is less than required for O_2 to reach peak abundance; (3) the O_2 emission arises within dense clumps with a small beam filling factor; or (4) the face-on depth into the Bar where O_2 reaches its peak abundance, which is density dependent, corresponds to a sky position different from that sampled by our Herschel beams
