Significance The replication stress response protein, ATR, is active in quiescent muscle stem cells in response to DNA:RNA hybrids, and this activity maintains quiescence by ensuring degradation of key cell-cycle transition factors by the E3 ubiquitin ligase cyclin F–SCF complex. This is critical for understanding how stem cells regulate a state of prolonged and reversible cell-cycle arrest and how genome integrity is maintained over time. Together, these studies offer a unique picture of a molecular mechanism controlling stem cell quiescence.