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The design of a system for accumulating counter-rotating beams of positrons and electrons, up to 0.1 A, in the CEA ring is discussed. Beams of this intensity are required if the CEA is to be used successfully as a colliding beam facility. Repeated, single turn, off-axis injection will be used at a repetition rate of 60 c/s. After each injection at 135 Mev, all particles will be accelerated to an energy in the 2 to 3 GeV range and then decelerated back to the injection energy. During the cycle, sufficient radiation damping of betatron and synchrotron oscillations is achieved to allow more particles to be injected without losing a large fraction of the particles injected in preceding cycles. Incoming particles are inflected by a septum magnet located off the equilibrium orbit while the path of the circulating beam is displaced towards the septum throughout the interval required for single turn injection. The current that can be accumulated depends on the number of particles injected per cycle and the lifetime of the circulating beam. Injection of a positron beam is more difficult than injection of an electron beam because of the smaller current and larger emittance associated with the source. When the horizontal phase space is used for injection, the half-life of the circulating beam is estimated to be 90 s if the average gas pressure is 5.108 torr. The positron source is expected to deliver 0.4 mA with an energy spread of 2.5 MeV aid an emittance of (2.