Eukaryotic chromosome ends are protected from illicit DNA joining by protein-DNA complexes called telomeres. In most studied organisms, telomeric DNA is composed of multiple short G-rich repeats that end in a single-stranded tail that is protected by the protein POT1. Mammalian POT1 binds two telomeric repeats as a monomer in a sequence-specific manner, and discriminates against RNA of telomeric sequence. While addressing the RNA discrimination properties of SpPot1, the POT1 homolog in Schizosaccharomyces pombe, we found an unanticipated ssDNA-binding mode in which two SpPot1 molecules bind an oligonucleotide containing two telomeric repeats. DNA binding seems to be achieved via binding of the most N-terminal OB domain of each monomer to each telomeric repeat. The SpPot1 dimer may have evolved to accommodate the heterogeneous spacers that occur between S. pombe telomeric repeats, and it also has implications for telomere architecture. We further show that the S. pombe telomeric protein Tpz1, like its mammalian homolog TPP1, increases the affinity of Pot1 for telomeric single-stranded DNA and enhances the discrimination of Pot1 against RNA.