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Antibodies are multimeric proteins capable of highly specific molecular recognition. The complementarity determining region 3 of the antibody variable heavy chain (CDRH3) often dominates antigen-binding specificity. Hence, it is a priority to design optimal antigen-specific CDRH3 to develop therapeutic antibodies. The combinatorial structure of CDRH3 sequences makes it impossible to query binding-affinity oracles exhaustively. Moreover, antibodies are expected to have high target specificity and developability. Here, we present AntBO, a combinatorial Bayesian optimization framework utilizing a CDRH3 trust region for an in silicodesign of antibodies with favorable developability scores. The in silicoexperiments on 159 antigens demonstrate that AntBOis a step toward practically viable in vitroantibody design. In under 200 calls to the oracle, AntBOsuggests antibodies outperforming the best binding sequence from 6.9 million experimentally obtained CDRH3s. Additionally, AntBOfinds very-high-affinity CDRH3 in only 38 protein designs while requiring no domain knowledge.