Spontaneous whole-genome duplication, or autodiploidization, is a common route to adaptation in experimental evolution of haploid budding yeast populations. The rate at which autodiploids fix in these populations appears to vary across strain backgrounds, but the genetic basis of these differences remains poorly characterized. Here, we show that the frequency of autodiploidization differs dramatically between two closely related laboratory strains of Saccharomyces cerevisiae, BY4741 and W303. To investigate the genetic basis of this difference, we crossed these strains to generate hundreds of unique F1 segregants and tested the tendency of each segregant to autodiplodize across hundreds of generations of laboratory evolution. We find that variants in the SSD1 gene are the primary genetic determinant of differences in autodiploidization. We then used multiple laboratory and wild strains of S. cerevisiae to show that clonal populations of strains with a functional copy of SSD1 autodiploidize more frequently in evolution experiments, while knocking out this gene or replacing it with the W303 allele reduces autodiploidization propensity across all genetic backgrounds tested. These results suggest a potential strategy for modifying rates of spontaneous whole-genome duplications in laboratory evolution experiments in haploid budding yeast. They may also have relevance to other settings in which eukaryotic genome stability plays an important role, such as biomanufacturing and the treatment of pathogenic fungal diseases and cancers.