IF the two gamma-quanta emitted on the annihilation of a slowly moving positron-electron pair both undergo a Compton scattering, in view of the fact that they are polarized in perpendicular directions, it is reasonable to expect that the scatter distributions will be correlated in azimuthal angle. Fig. 1 shows the expected state of affairs. It has recently been proposed by R. C. Hanna to detect this effect, using counters to observe coincidences of the scattered quanta. With the view of assisting the design of the experiment, we have calculated the magnitude of the effect to be expected. We find that the differential double cross-section is given by where r0 is the classical electron radius, and dΩ1, dΩ2 are elements of solid angle for quanta 1 and 2 respectively.