The spike protein S of SARS coronavirus 2 (SARS-CoV-2) binds ACE2 on host cells to initiate entry, and soluble ACE2 is a therapeutic candidate that neutralizes infection by acting as a decoy. Using deep mutagenesis, mutations in ACE2 that increase S binding are found across the interaction surface, in the N90-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 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.