The adoptive transfer of T cells expressing anti-CD19 chimeric antigen receptors (CARs) has shown remarkable curative potential against advanced B-cell malignancies, but multiple trials have also reported patient relapses due to the emergence of CD19-negative leukemic cells. Here, we report the design and optimization of single-chain, bispecific CARs that trigger robust cytotoxicity against target cells expressing either CD19 or CD20, two clinically validated targets for B-cell malignancies. We determined the structural parameters required for efficient dual-antigen recognition, and we demonstrate that optimized bispecific CARs can control both wild-type B-cell lymphoma and CD19(-) mutants with equal efficiency in vivo To our knowledge, this is the first bispecific CAR capable of preventing antigen escape by performing true OR-gate signal computation on a clinically relevant pair of tumor-associated antigens. The CD19-OR-CD20 CAR is fully compatible with existing T-cell manufacturing procedures and implementable by current clinical protocols. These results present an effective solution to the challenge of antigen escape in CD19 CAR T-cell therapy, and they highlight the utility of structure-based rational design in the development of receptors with higher-level complexity.