Chronic infections are difficult to treat with antibiotics but are caused primarily by drug-sensitive pathogens. Dormant persister cells that are tolerant to killing by antibiotics are responsible for this apparent paradox. Persisters are phenotypic variants of normal cells and pathways leading to dormancy are redundant, making it challenging to develop anti-persister compounds. Biofilms shield persisters from the immune system, suggesting that an antibiotic for treating a chronic infection should be able to eradicate the infection on its own. We reasoned that a compound capable of corrupting a target in dormant cells will kill persisters. The acyldepsipeptide antibiotic (ADEP4) has been shown to activate the ClpP protease, resulting in death of growing cells. Here we show that ADEP4-activated ClpP becomes a fairly nonspecific protease and kills persisters by degrading over 400 proteins, forcing cells to self-digest. Null mutants of clpP arise with high probability, but combining ADEP4 with rifampicin produced complete eradication of Staphylococcus aureus biofilms in vitro and in a mouse model of a chronic infection. Our findings indicate a general principle for killing dormant cells—activation and corruption of a target, rather than conventional inhibition. Eradication of a biofilm in an animal model by activating a protease suggests a realistic path towards developing therapies to treat chronic infections.