The proteasome is the main ATP-dependent protease in eukaryotic cells and controls the concentration of many regulatory proteins in the cytosol and nucleus. Proteins are targeted to the proteasome by the covalent attachment of polyubiquitin chains. The ubiquitin modification serves as the proteasome recognition element but by itself is not sufficient for efficient degradation of folded proteins. We report that proteolysis of tightly folded proteins is accelerated greatly when an unstructured region is attached to the substrate. The unstructured region serves as the initiation site for degradation and is hydrolyzed first, after which the rest of the protein is digested sequentially. These results identify the initiation site as a novel component of the targeting signal, which is required to engage the proteasome unfolding machinery efficiently. The proteasome degrades a substrate by first binding to its ubiquitin modification and then initiating unfolding at an unstructured region.