Prime editing enables the precise modification of genomes through reverse transcription of template sequences appended to the 3' ends of CRISPR-Cas guide RNAs. To identify cellular determinants of prime editing, we developed scalable prime editing reporters and performed genome-scale CRISPR-interference screens. From these screens, a single factor emerged as the strongest mediator of prime editing: the small RNA-binding exonuclease protection factor La. Further investigation revealed that La promotes prime editing across approaches (PE2, PE3, PE4 and PE5), edit types (substitutions, insertions and deletions), endogenous loci and cell types but has no consistent effect on genome-editing approaches that rely on standard, unextended guide RNAs. Previous work has shown that La binds polyuridine tracts at the 3' ends of RNA polymerase III transcripts. We found that La functionally interacts with the 3' ends of polyuridylated prime editing guide RNAs (pegRNAs). Guided by these results, we developed a prime editor protein (PE7) fused to the RNA-binding, N-terminal domain of La. This editor improved prime editing with expressed pegRNAs and engineered pegRNAs (epegRNAs), as well as with synthetic pegRNAs optimized for La binding. Together, our results provide key insights into how prime editing components interact with the cellular environment and suggest general strategies for stabilizing exogenous small RNAs therein.