This article uses a socio-anthropological framework to explore the stigmas around interactions with children born with congenital Zika syndrome caused by the Zika virus epidemic in two Brazilian municipalities. Semi-structured interviews were conducted with parents and other relatives. We reflected on the search for meaning when having a baby with unexpected body marks, the moral suffering, the societal ableism, the burden of care, and the need for support networks. We concluded that public policies, especially social policies (health, education, and social assistance), are essential for compensatory mechanisms, recognition, and social inclusion of these children and their families.