The Ku complex binds non-specifically to DNA breaks and ensures repair via NHEJ. However, Ku is also known to bind directly to telomeric DNA ends and its presence there is associated with telomere capping, but avoiding NHEJ. How the complex discriminates between a DNA break and a telomeric extremity remains unknown. Our results using a tagged Ku complex, or a chromosome end capturing method, in budding yeast show that yKu association with telomeres can occur at sites distant from the physical end, on sub-telomeric elements, as well as on interstitial telomeric repeats. Consistent with previous studies, our results also show that yKu associates with telomeres in two distinct and independent ways: either via protein-protein interactions between Yku80 and Sir4 or via direct DNA binding. Importantly, yKu associates with the new sites reported here via both modes. Therefore, in sir4Δ cells, telomere bound yKu molecules must have loaded from a DNA-end near the transition of non-telomeric to telomeric repeat sequences. Such ends may have been one sided DNA breaks that occur as a consequence of stalled replication forks on or near telomeric repeat DNA. Altogether, the results predict a new model for yKu function at telomeres that involves yKu binding at one-sided DNA breaks caused by replication stalling. On telomere proximal chromatin, this binding is not followed by initiation of non-homologous end-joining, but rather by break-induced replication or repeat elongation by telomerase. After repair, the yKu-distal portion of telomeres is bound by Rap1, which in turn reduces the potential for yKu to mediate NHEJ. These results thus propose a solution to a long-standing conundrum, namely how to accommodate the apparently conflicting functions of Ku on telomeres.