The case has raised ethical questions about how well a child could cope with such unusual family circumstances where it would be its grandfather's genetic child, and a halfbrother or half-sister to man whom it would be calling its father.
The Harley Street doctors treating the couple are not disclosing their identity. They, however, have revealed that the pair opted for donor sperm after IVF treatment with the husband's own sperm failed.
According to the doctors, the couple chose the husband's father as a sperm donor because they wanted their child to be genetically similar to both families.
"I've certainly never come across a case like this before. But advancements in fertility treatment have overcome a lot of taboos in science which means that people are prepared to consider all sorts of options," the Daily Mail quoted him as saying.
"Obviously, the wife's mother-inlaw also had to be included in all of the conversations but she has no objections. Society has also changed its perceptions of what is and what is not acceptable. In this case, keeping the genetic identity of the child similar to their own was a huge factor. The husband does not have a brother, which is why he chose his own father to assist," he added.
Kamal Ahuja, the clinic's scientific director, says that before giving the green light to this procedure, the authorities had considered ethical questions attached to this case.
"We spent many, many months deliberating this case and discussed it with our ethics committee and with counsellors and have come to the conclusion that they shouldn't have been denied treatment," he said.
Although it is yet unclear whether the couple want to tell the child about its parentage, it has been revealed that the child will be able to track down its biological father on turning 18.
Meanwhile, critics have cautioned that the child born through this procedure may might face major identity issues.
"The reproductive whims of parents to do some deliberate and unnecessary social engineering should not be put before the welfare of the child. Just because you can do something, doesn't mean you should," Philippa Taylor of Care, a Christian charity, said.
The Human Fertilisation and Embryology Authority has clearly stated that donations from family members are legal and relatively common