In budding yeast, Cdc13, Stn1, and Ten1 form the telomere-binding heterotrimer CST complex. Here we investigate the role of Cdc13/CST in maintaining genome stability by using a Chr VII disome system that can generate recombinants, chromosome loss, and enigmatic unstable chromosomes. In cells expressing a temperature sensitive CDC13 allele, cdc13(F684S), unstable chromosomes frequently arise from problems in or near a telomere. We found that, when Cdc13 is defective, passage through S phase causes Exo1-dependent ssDNA and unstable chromosomes that are then the source for additional chromosome instability events (e.g. recombinants, chromosome truncations, dicentrics, and/or chromosome loss). We observed that genome instability arises from a defect in Cdc13's function during DNA replication, not Cdc13's putative post-replication telomere capping function. The molecular nature of the initial unstable chromosomes formed by a Cdc13-defect involves ssDNA and does not involve homologous recombination nor non-homologous end joining; we speculate the original unstable chromosome may be a one-ended double strand break. This system defines a link between Cdc13's function during DNA replication and genome stability in the form of unstable chromosomes, that then progress to form other chromosome changes. Author summary Eukaryotic chromosomes are linear molecules with specialized end structures called telomeres. Telomeres contain both unique repetitive DNA sequences and specialized proteins that solve several biological problems by differentiating chromosomal ends from internal breaks, thus preventing chromosome instability. When telomeres are defective, the entire chromosome can become unstable and change, causing mutations and pathology (cancer, aging, etc.). Here we study how a defect in specific telomere proteins causes chromosomal rearrangements, using the model organism Saccharomyces cerevisiae (budding or brewer's yeast). We find that when specific telomere proteins are defective, errors in DNA replication generate a type of damage that likely involves extensive single-stranded DNA that forms inherently unstable chromosomes, subject to many subsequent instances of instability (e.g. allelic recombinants, chromosome loss, truncations, dicentrics). The telomere protein Cdc13 is part of a protein complex called CST that is conserved in most organisms including mammalian cells. The technical capacity of studies in budding yeast allow a detailed, real-time examination of how telomere defects compromise chromosome stability in a single cell cycle, generating lessons likely relevant to how human telomeres keep human chromosomes stable.