The design and optimization of antibodies, important therapeutic agents, requires an intricate balance across multiple properties. A primary challenge in optimization is ensuring that introduced sequence mutations do not disrupt the antibody structure or its target binding mode. Protein inverse folding models, which predict diverse sequences that fold into the same structure, are promising for maintaining structural integrity during optimization. Here we present AntiFold, an inverse folding model developed for solved and predicted antibody structures, based on the ESM-IF1 model. AntiFold achieves large gains in performance versus existing inverse folding models on sequence recovery, across all antibody complementarity determining regions (CDRs) and framework regions. AntiFold-generated sequences show high structural agreement between predicted and experimental structures. The tool efficiently samples hundreds of antibody structures per minute, providing a scalable solution for antibody design. AntiFold is freely available for academic use as a downloadable package at: https://opig.stats.ox.ac.uk/data/downloads/AntiFold