Photodissociation regions (PDRs), where UV radiation dominates the energetics and chemistry of the neutral gas, contain most of the mass in the dense interstellar medium of our Galaxy. Observations of H 2 rotational and rovibrational lines reveal that PDRs contain unexpectedly large amounts of very warm (400-700 K) molecular gas. Theoretical models have difficulty explaining the existence of so much warm gas. Possible problems include errors in the heating and cooling functions or in the formation rate for H 2. To date, observations of H 2 rotational lines smear out the structure of the PDR. Only by resolving the hottest layers of H 2 can one test the predictions and assumptions of current models. Using the Texas Echelon Cross Echelle Spectrograph (TEXES) we mapped emission in the H 2 v= 0-0 S (1) and S (2) lines toward the Orion Bar PDR at 2''resolution. We also observed H 2 v= 0-0 S (4) at selected points toward the front of the PDR. Our maps cover a 12''by 40''region of the bar where H 2 rovibrational lines are bright. The distributions of H 2 0-0 S (1), 0-0 S (2), and 1-0 S (1) line emission agree in remarkable detail. The high spatial resolution (0.002 pc) of our observations allows us to probe the distribution of warm gas in the Orion Bar to a distance approaching the scale length for FUV photon absorption. We use these new observational results to set parameters for the PDR models described in a companion paper in preparation by Draine et al. The best-fit model can account for the separation of the H 2 emission from the ionization front and the intensities of the ground-state rotational lines, as well as the 1-0 S (1) and 2-1 S (1) lines. This model requires significant adjustments to the commonly used values for the dust UV attenuation cross section and the photoelectric heating rate.