We describe a general assay designed to detect mutants of yeast that are defective for any of several aspects of telomere function. Using this assay, we have isolated a mutant that displays a progressive decrease in telomere length as well as an increased frequency of chromosome loss. This mutation defines a new gene, designated EST1 (for ever shorter telomeres). Null alleles of EST1 are not immediately inviable; instead, they have a senescence phenotype, due to the gradual loss of sequences essential for telomere function, leading to a progressive decrease in chromosomal stability and subsequent cell death.