The overexpression and misfolding of viral proteins in the endoplasmic reticulum (ER) may cause cellular stress, thereby inducing cytoprotective proteostatic host responses involving phosphorylation of eIFa. Here, we show the positive-strand RNA virus responsible for infectious hepatitis adopts a stress-resistant, phospho-eIF2a independent mechanism of translation that ensures synthesis of its viral proteins within the infected liver. Cap-independent hepatovirus translation and productive infection in vivo requires PDAP1, a small phosphoprotein with previously unrecognized eIF4E-binding activity. We show PDAP1 interacts also with eIF1A, and is essential for translation of stress-resistant host mRNAs that evade the proteostatic response to ER stress.