The 2015–2016 outbreak of Zika virus (ZIKV) disease has affected many countries and is a major public health concern. ZIKV is associated with fetal microcephaly and neurological complications, and countermeasures are needed to treat and prevent ZIKV infection. We report the isolation of 13 specific human monoclonal antibodies from a single patient infected with ZIKV. Two of the isolated antibodies (Z23 and Z3L1) demonstrated potent ZIKV-specific neutralization in vitro without binding or neutralizing activity against strains 1 to 4 of dengue virus, the closest relative to ZIKV. These two antibodies provided postexposure protection to mice in vivo. Structural studies revealed that Z23 and Z3L1 bound to tertiary epitopes in envelope protein domain I, II, or III, indicating potential targets for ZIKV-specific therapy. Our results suggest the potential of antibody-based therapeutics and provide a structure-based rationale for the design of future ZIKV-specific vaccines.