When telomeres shorten, they are thought to lose their protective caps at a critical short length, activating the DNA damage response and recruiting DNA damage repair activities that would degrade, fuse or recombine dysfunctional telomeres. However, the structure(s) of short and dysfunctional telomeres, which respectively trigger permanent replicative senescence and/or potentially promote genome instability, remain unclear. To define the structure of telomeres at the point of dysfunction and the fate of cells carrying them, we have developed a system called FinalCut in Saccharomyces cerevisiae to induce a single telomere of defined length in cells in which we can conditionally inactivate telomerase. This allows 1/ the structural analysis of the shortest telomere in cells, 2/ the monitoring of the pace and the number of consecutive cell cycles from telomerase inactivation to cell death with our microfluidic system, and 3/ the combination of our single cell - single telomere resolved data with a mathematical model of senescence in budding yeast. Our results show that a very small fraction of cells can enter senescence in a probabilistic manner, as cells proliferate in the absence of telomerase. However, for the large majority of cells, when the shortest telomere reaches a critical threshold length, cells immediately stop proliferation. This is consistent with the model that the first telomere reaching a short threshold length is sufficient to abruptly stop cell divisions in a large majority of cells. The structure of this critically short telomere will be discussed.