The SARS-CoV-2 Omicron variant exhibits very high levels of transmission, pronounced resistance to authorized therapeutic human monoclonal antibodies and reduced sensitivity to vaccine-induced immunity. Here we describe P2G3, a human monoclonal antibody (mAb) isolated from a previously infected and vaccinated donor, which displays picomolar-range neutralizing activity against Omicron BA.1, BA.1.1, BA.2 and all other current variants, and is thus markedly more potent than all authorized or clinically advanced anti-SARS-CoV-2 mAbs. Structural characterization of P2G3 Fab in complex with the Omicron Spike demonstrates unique binding properties to both down and up spike trimer conformations at an epitope that partially overlaps with the receptor-binding domain (RBD), yet is distinct from those bound by all other characterized mAbs. This distinct epitope and angle of attack allows P2G3 to overcome all the Omicron mutations abolishing or impairing neutralization by other anti-SARS-COV-2 mAbs, and P2G3 accordingly confers complete prophylactic protection in the SARS-CoV-2 Omicron monkey challenge model. Finally, although we could isolate in vitro SARS-CoV2 mutants escaping neutralization by P2G3 or by P5C3, a previously described broadly active Class 1 mAb, we found these viruses to be lowly infectious and their key mutations extremely rare in the wild, and we could demonstrate that P2G3/P5C3 efficiently cross-neutralized one another's escapees. We conclude that this combination of mAbs has great prospects in both the prophylactic and therapeutic settings to protect from Omicron and other VOCs.