We mapped the quiescent emission from the 807 GHz J = 7 --> 6 transiti
on of CO in Orion along a strip in R.A. extending from 0.7 pc west to
1.2 pc east of theta1C Orionis. The lines arise in warm gas with tempe
ratures greater than 40 K. The line brightness temperature is greater
than 160 K in the direction of theta1C, more than twice the dust tempe
rature, and still exceeds 35 K more than a parsec east of theta1C. The
lines are narrow, with a maximum velocity width of 7 km s-1 near thet
a1C and decreasing to 1.5-3 km s-1 at the map boundaries. The density
of the emitting gas is greater than 10(4) cm-3 and the column density
exceeds 10(21) cm-2. The correlation of the bright, narrow CO(7 --> 6)
lines with 158 mum [C II] emission suggests that over the entire regi
on mapped, the narrow CO lines arise in warm photodissociation regions
excited by ultraviolet (UV) photons from the Trapezium cluster. Altho
ugh the Trapezium stars lie in front of the Orion A molecular cloud, n
ot all of the warm gas is at the cloud surface. To the east of theta1C
the CO(7 --> 6) lines split into two velocity components (also seen i
n J = 2 --> 1 (CO)-C-13 emission) which persist over several arcminute
s. Since only one of these components can be on the surface, the other
must arise from a dense, UV-illuminated clump or filament within the
molecular cloud. Comparison of the quiescent CO(7 --> 6) emission to C
O(7 --> 6) observed in a cross map of the energetic Orion KL outflow s
hows that the luminosity of shock-excited CO(7 --> 6) emission in Orio
n is only a few percent of the luminosity of the widespread quiescent
CO(7 --> 6) emission.