We discovered a simple regime where a near-critical plasma irradiated by a laser of experimentally available intensity can self-organize to produce positrons and accelerate them to ultra-relativistic energies. The laser pulse piles up electrons at its leading edge, producing a strong longitudinal plasma electric field. The field creates a moving gamma-ray collider that generates positrons via the linear Breit-Wheeler process -- annihilation of two gamma-rays into an electron-positron pair. At the same time, the plasma field, rather than the laser, serves as an accelerator for the positrons. The discovery of positron acceleration was enabled by a first-of-its-kind kinetic simulation that generates pairs via photon-photon collisions. Using available laser intensities of $10^{22}$$\ $$\rm W/cm^2$, the discovered regime can generate a GeV positron beam with divergence angle of $\sim10^{\circ}$ and total charge of 0.1$\ $pC. The result paves the way to experimental observation of the linear Breit-Wheeler process and to applications requiring positron beams.
