Confirming an earlier, although preliminary result, we present for the first time firm evidence for the spatial convergence of a cosmological dipole (i.e., peculiar acceleration). The objects used in the analysis are clusters of galaxies from the Abell and ACO catalogs. The cluster dipole increases with distance up to approximately 180h-1 Mpc, then bends over and stays flat well within the depth of the sample, while the monopole continues to grow linearly with distance up to the maximum sample depth, 300h-1 Mpc, where it reaches a level which is greater-than-or-similar-to 5 times the dipole level. The still-growing monopole proves that the convergence of the dipole is real and is not due to depth incompleteness of the sample. The behavior of two such quantities is the one expected in an asymptotically homogeneous and isotropic universe and gives further three-dimensional (all-sky) support to a FRW model, besides the two-dimensional one from cosmic backgrounds. After suitable corrections for the differences in the two catalogs of clusters of galaxies (Abell and ACO) limited at (b) > 20-degrees, the asymptotic dipole direction appears to be stable pointing in a direction almost-equal-to 25-degrees off the CMB dipole apex. By a posteriori pushing the limit at (b) > 10-degrees to include the effects of the Perseus cluster, we find that, after correction for the missing Galactic strip, the dipole direction is within only 6-degrees-10-degrees of the CMB apex. An analysis of the behavior of the dipole as a function of distance suggests that a nonnegligible fraction of the peculiar acceleration responsible for the LG motion comes from greater depths than concluded from previous studies of IRAS and optical catalogs of galaxies, with possible important consequences on the viable cosmological scenarios. From these data we estimate that less-than-or-similar-to one-third of the acceleration comes from d greater-than-or-equal-to 60h-1 Mpc. The same data can also be used to estimate the density parameter OMEGA-0. After rescaling the cluster dipole to monopole ratio by the value of the cluster-to-galaxy bias derived from the correlation functions (b(cg) almost-equal-to 3), we obtain OMEGA-0, = 0.2-0.4 for no or mild bias of galaxies (b(g-rho) congruent-to 1-1.5). An OMEGA-0 = 1 universe could still be accommodated by requiring a very large bias of galaxies with respect to matter, b(g-rho) congruent-to 2.7-3, which is, however, not supported by several large-scale observations.