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10 pages, 7 figures.-- PACS nrs.: 57.10.Pq; 75.30.-m; 75.30.Gw; 78.70.Dm.-- Published online Oct 22, 2005., We present a combined experimental and theoretical study of the magnetization of one-dimensional atomic cobalt chains deposited on a platinum surface. We discuss the intrinsic magnetization parameters derived by X-ray magnetic circular dichroism measurements and the observation of ferromagnetic order in one dimension in connection with the presence of strong, dimensionality-dependent anisotropy energy barriers of magnetocrystalline origin. An explicit transfer matrix formalism is developed to treat atomic chains of finite length within the anisotropic Heisenberg model. This model allows us to fit the experimental magnetization curves of cobalt monatomic chains, measured parallel to the easy and hard axes, and provides values of the exchange coupling parameter and the magnetic anisotropy energy consistent with those reported in the literature. The analysis of the spin–spin correlation as a function of temperature provides further insight into the tendency to magnetic order in finite-sized one-dimensional systems., We would like to acknowledge the contribution to the experiments reported in this work by A. Dallmeyer, K. Maiti, M.C. Malagoli, and W. Eberhardt (Forschungszentrum Jülich), P. Ohresser, S.S. Dhesi, N.B. Brookes, and K. Larsson (beamline ID12B at the ESRF), and K. Kern (EPF Lausanne and MPI Stuttgart). A. Vindigni, A. Rettori, and M.G. Pini acknowledge financial support from the Italian MIUR (projects FISR ‘Nanotecnologie per dispositivi di memoria’ and FIRB ‘Microsistemi e nanomateriali magnetici’).