Hst3 is the histone deacetylase that removes histone H3K56 acetylation. H3K56 acetylation is a cell-cycle- and damage-regulated chromatin marker, and proper regulation of H3K56 acetylation is important for replication, genomic stability, chromatin assembly, and the response to and recovery from DNA damage. Understanding the regulation of enzymes that regulate H3K56 acetylation is of great interest, because the loss of H3K56 acetylation leads to genomic instability. HST3 is controlled at both the transcriptional and posttranscriptional level. Here, we show that Hst3 is targeted for turnover by the ubiquitin ligase SCF(Cdc4) after phosphorylation of a multisite degron. In addition, we find that Hst3 turnover increases in response to replication stress in a Rad53-dependent way. Turnover of Hst3 is promoted by Mck1 activity in both conditions. The Hst3 degron contains two canonical Cdc4 phospho-degrons, and the phosphorylation of each of these is required for efficient turnover both in an unperturbed cell cycle and in response to replication stress.