Junior health minister Lord Hunt told the House of Lords that the government had been asked to keep the probe secret by a judge.
A court order bars the identification of anyone involved in the case, and the clinic responsible for the mix-up.
However, Lord Hunt revealed the progress of the investigation in question time exchanges on Thursday because, he said, he did not wish to mislead the House if asked directly about the case.
He said: "We have arranged for a very thorough independent investigation to be carried out.
"At last week's court hearing the judge directed the government that it should not announce it had set up an investigation.
"This was because she was concerned to keep discussion of this case to a minimum to protect the families and children concerned."
Question marks
Opposition spokesman Earl Howe said the "disturbing" story gave rise to "serious concerns" about the role of the Human Fertilisation and Embryology Authority - which is expected to be criticised heavily in a report by the House of Commons Science and Technology Committee on Thursday.
He said: "What confidence can people have that procedures followed in IVF clinics are proof against error?"
The exact circumstances of the mix-up have not yet been revealed, but could involve either the egg from the white mother being fertilised by sperm from the black patient, or an embryo created from sperm and eggs from the black couple being implanted into the wrong woman.
It is not known if the black couple is aware that the mistake has been made.