The overexpression and misfolding of viral proteins in the endoplasmic reticulum (ER) may cause cellular stress, thereby inducing a cytoprotective, proteostatic host response involving phosphorylation of eukaryotic translation initiation factor 2 subunit alpha (eIF2α). Here, we show that hepatitis A virus, a positive-strand RNA virus responsible for infectious hepatitis, adopts a stress-resistant, eIF2α-independent mechanism of translation to ensure the synthesis of viral proteins within the infected liver. Cap-independent translation directed by the hepatovirus internal ribosome entry site and productive hepatovirus infection of mice both require platelet-derived growth factor subunit A (PDGFA)–associated protein 1 (PDAP1), a small phosphoprotein of unknown function with eIF4E-binding activity. PDAP1 also interacts with eIF1A and is essential for translating stress-resistant host messenger RNAs that evade the proteostatic response to ER stress and that encode proteins promoting the survival of stressed cells.