Therapeutic antibodies are an important and rapidly growing drug modality. However, the design and discovery of early-stage antibody therapeutics remain a time and cost-intensive endeavor. Here we present an end-to-end Bayesian, language model-based method for designing large and diverse libraries of high-affinity single-chain variable fragments (scFvs) that are then empirically measured. In a head-to-head comparison with a directed evolution approach, we show that the best scFv generated from our method represents a 28.7-fold improvement in binding over the best scFv from the directed evolution. Additionally, 99% of designed scFvs in our most successful library are improvements over the initial candidate scFv. By comparing a library’s predicted success to actual measurements, we demonstrate our method’s ability to explore tradeoffs between library success and diversity. Results of our work highlight the significant impact machine learning models can have on scFv development. We expect our method to be broadly applicable and provide value to other protein engineering tasks.