The ends of eukaryotic chromosomes are protected by specialized telomere chromatin structures. Rap1 and Cdc13 are essential for the formation of functional telomere chromatin in budding yeast by binding to the double-stranded part and the single-stranded 3' overhang, respectively. We analyzed the binding properties of Saccharomyces castellii Rap1 and Cdc13 to partially single-stranded oligonucleotides, mimicking the junction of the double- and single-stranded DNA (ds-ss junction) at telomeres. We determined the optimal and the minimal DNA setup for a simultaneous binding of Rap1 and Cdc13 at the ds-ss junction. Remarkably, Rap1 is able to bind to a partially single-stranded binding site spanning the ds-ss junction. The binding over the ds-ss junction is anchored in a single double-stranded hemi-site and is stabilized by a sequence-independent interaction of Rap1 with the single-stranded 3' overhang. Thus, Rap1 is able to switch between a sequence-specific and a non-specific binding mode of one hemi-site. At a ds-ss junction configuration where the two binding sites partially overlap, Rap1 and Cdc13 are competing for the binding. These results shed light on the end protection mechanisms, and suggest that Rap1 and Cdc13 act together to ensure the protection of both the 3' and the 5' DNA ends at telomeres.