The spike (S) protein of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) binds angiotensin-converting enzyme 2 (ACE2) on host cells to initiate entry, and soluble ACE2 is a therapeutic candidate that neutralizes infection by acting as a decoy. By using deep mutagenesis, mutations in ACE2 that increase S binding are found across the interaction surface, in the asparagine 90-glycosylation motif and at buried sites. The mutational landscape provides a blueprint for understanding the specificity of the interaction between ACE2 and S and for engineering high-affinity decoy receptors. Combining mutations gives ACE2 variants with affinities that rival those of monoclonal antibodies. A stable dimeric variant shows potent SARS-CoV-2 and -1 neutralization in vitro. The engineered receptor is catalytically active, and its close similarity with the native receptor may limit the potential for viral escape.