We report on a measurement of the branching ratio of the rare decay omega -
-> eta gamma relative to the well known decay omega --> pi(0)gamma. The ome
ga's are produced in <p(p)over bar> --> eta omega and <p(p)over bar> --> pi
(0)omega. Eigenstate mixing and interference effects. of the wand rho(0) ar
e taken into account, as well as coherent interference with the background.
We find evidence for the non-resonant annihilation channel B(<p(p)over bar
> --> eta eta gamma) = (3.5 +/- 1.3) x 10(-5) and limit the value of B(omeg
a --> eta gamma) to the range of (0.7 to 5.5) x 10(-4) depending on the deg
ree of coherence with the background.