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Starting from the observed distribution of Abell/ACO galaxy clusters in redshift space, we use a two-step procedure to recover their distances and peculiar velocities. After statistically correcting for the unobserved cluster distribution in the zone of avoidance and also for a smooth absorption at higher latitudes, we use a dynamical iterative algorithm, based on that published by Strauss & Davis in 1988, to recover the real-space duster positions by minimizing the redshift-space distortions. The whole procedure assumes that clusters trace the mass, that peculiar velocities are caused by gravity, and that linear perturbation theory applies. The amplitude of the cluster dipole measured in real space turns out to be similar to 23% less than that measured in redshift space. In both cases the dipole direction is aligned with the cosmic microwave background dipole within similar to 10 degrees, taking into account the Virgocentric infall component of the Local Group motion. Using linear theory we obtain beta(c)(=Omega(0)(0.6)/b(c)) approximate to 0.21(+/-0.03), where the uncertainty is due to observational errors and limitations in the reconstruction procedure, while the intrinsic cosmological variance amounts to similar to 0.07. This beta(c) value implies that for a cluster-mass bias parameter of b(c) less than or similar to 5, a flat universe is not excluded, contrary to previous cluster-dipole z-space analysis. A more stringent determination of beta(c) will be obtained from the analysis of the peculiar velocity field in a forthcoming paper

Reconstructing Positions and Peculiar Velocities of Galaxy Clusters within 25,000 Kilometers per Second: The Cluster Real Space Dipole