The parents claim their child is perfectly healthy and that they should be able to decide what is best for her. But Camden Council has asked the High Court in London to allow it to override their wishes because the mother is HIV-positive.
The landmark case, being heard in private in the Family Division of the High Court, is expected to have huge implications for the rights of parents to make decisions about the health of their children.
The Official Solicitor, Mr Laurence Oates, has been asked to represent the interests of the child and report to the court.
Mr Justice Wilson, who is hearing the matter, said in view of the "very legitimate public interest" he would make his ruling public next week. This was so that the public could be aware of how he had resolved the issue and so that it could continue to debate the "difficult matters" raised.
Social workers in Camden, north London, made the application under the 1989 Children's Act, saying HIV is a deadly infection which could kill the girl.
If found to be HIV-positive, the child could be treated with drugs to combat the virus. If she tests negative, her mother could be urged to stop breast-feeding in a bid to stop the baby becoming infected.
The child's father, an alternative healthcare practitioner, has tested negative for the virus. The mother, who was found to be HIV-positive in 1990, believes scientists are mistaken in seeing HIV as the sole cause of Aids.
But the couple's GP and experts at Great Ormond Street Hospital urged them to have their daughter tested.
The father has been reported as saying: "This is yet another example of the state trying to control every aspect of family life. Making the decision to have a child was not something that we took lightly."
He added: "This baby is loved and cared for beyond reproach. The council is acting in both ignorance and fear."
A spokesman for Camden Council, which has received the public backing of Aids charities, would only confirm that the case was going ahead.Reuse content