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We have studied the magnetic ordering of terbium overlayers on Cu(100) and Ni(111), using angle-resolved photoemission. The 5p3/2 to 5p1/2 shallow-core-level branching ratios in different photoemission geometries provide a measure of the magnetic ordering in rare-earth-metal overlayers as a result of final-state effects in photoemission. We find that ferromagnetic substrates order paramagnetic terbium overlayers. This induced magnetic ordering is not a crystal-field effect and can be modeled by Ginzburg-Landau theory. Application of Ginzburg-Landau theory to our results suggests that the correlation length of paramagnetic terbium κ-1 is between 2.5 and 3.5 Å. Reversible increases in the extent of magnetic ordering at temperatures below the Tb Curie temperature are observed for terbium overlayers on both Cu(100) and Ni(111).