She grew from an embryo screened to ensure it did not contain the faulty BRCA 1 gene, carried by 37,000 women in the UK, which passes the risk of breast cancer down generations.
University College London said the the mother and her little girl were doing "very well".
Doctors hope that the child will be able to break a family history of the disease.
The birth could also signal new hope for other families which carry the gene, known to radically increase the chance of developing breast cancer, which kills around 15,000 women in Britain every year.
Announcing her birth, Paul Serhal, medical director of the Assisted Conception Unit at the hospital, said: "This little girl will not face the spectre of developing this genetic form of breast cancer or ovarian cancer in her adult life.
"The parents will have been spared the risk of inflicting this disease on their daughter.
"The lasting legacy is the eradication of the transmission of this form of cancer that has blighted these families for generations."
Doctors at University College Hospital in London used a new test to screen a 27-year-old woman's embryos and specifically select those which did not carry the gene.
The women and her husband went through the procedure because of a high likelihood that they would pass on the inherited susceptibility to the disease.
He is a carrier of the gene and three generations of his family have been diagnosed with breast cancer while in their twenties, including his mother, sister, cousin and his grandmother.
Women who carry the gene have a between 50 and 80 per cent chance of going on to develop breast cancer at some point in their life.
In the past women who have been identified as carrying the gene have undergone radical operations including to remove their breasts to avoid contracting the cancer.
Doctors screen for the gene by examining a single cell removed from the embryo when it is around three days old.
From this they can tell which of a woman's embryos do not carry the breast cancer gene.
These are then implanted into the womb using IVF treatment.
Around 1,000 babies are already thought to have been born through this method which doctors have used to identify cystic fibrosis and a type of eye cancer.
In Britain eight fertility clinics offer couples the treatment, called Preimplantation Genetic Diagnosis (PGD) , if they have inherited diseases in the family.
But the procedure has proved controversial because otherwise healthy embryos are discarded because they carry a certain gene.