Sir Elton John, 68, and his husband David Furnish, 52, meanwhile, have two sons via a surrogate, whom clothes designers Dolce and Gabbana controversially described as "synthetic" earlier this month.
But it's not just celebrities who are making surrogacy fashionable. Recent figures from the Children and Family Court Advisory and Support Service (Cafcass) show that record numbers of UK babies are being born to surrogate parents - 167 last year, up from 47 in 2007. Like the Griffiths, many couples go to countries where, unlike in the UK, commercial surrogacy is legal. Data released earlier this month show that in the past three years more than 1,000 hopeful couples have travelled to a total of 57 such countries.
The process is not without controversy. Last month Thailand banned commercial surrogacy for foreign couples following the case of Baby Gammy, the twin boy an Australian couple allegedly left with his Thai surrogate mother following a diagnosis of Down's syndrome. The Foreign Office warns couples about the lack of international regulation and the risk of commercial exploitation of surrogate mothers from poorer countries.
As in Caroline's case, surrogacy also provokes strong views about very late motherhood. "There is a doubt in my mind that Grace may be lacking an able-bodied and able-minded mother when she needs me as a teenager," she admits. "But at the moment I have lots of energy to look after her."
The couple, from London, met in February 2000 when Nigel, a chartered surveyor, was 31 and Caroline 42. "When I was younger it was difficult to have a successful career and children," says Caroline, who owns an HR company. "When I met Nigel I hoped it would happen naturally."
By the time Caroline was 45 and still not pregnant, she and Nigel decided to pursue IVF, although by this time Caroline could not use her own eggs.
"A woman has produced her best eggs by her early thirties and by 43 they are unlikely to work," explains Dr Susan Ingamells, a consultant in reproductive medicine at Wessex Fertility Centre in Southampton.
"It was upsetting," says Caroline. "But the consultant said that if I used a donor egg I could still become a mother."
Over the next decade the couple underwent nine cycles of IVF treatment, using Nigel's sperm and donor eggs, at private clinics in Australia, Spain, London and Turkey. Each attempt failed, leaving Caroline desolate. "After every cycle there was an incredible sense of emptiness," she recalls. At around pounds 5,000 per cycle of IVF, the process also proved costly.
By the time she was 55, Caroline finally accepted that she would need to use a surrogate if she wanted to become a mother. "I reconciled myself to the fact that carrying a child would have been unsafe," she says.
She and Nigel discovered that finding a surrogate in the UK could be difficult and protracted. "Although surrogacy in the UK isn't illegal, there is no contract you can enter into with the surrogate mother and no access to the agency services that are plentiful abroad," says Natalie Gamble, a surrogacy lawyer based in London. Surrogate mothers here can only receive expenses, as opposed to formal payment, for their services, and English law treats the surrogate as the legal mother until a parental order is signed. "If they decide they want to keep the baby there is nothing you can do," says Caroline. "We thought it better to go to a country where surrogacy was properly regulated and we could be surer of an end result."
Research led them to visit the New Life Clinic in Tbilisi, capital of Georgia, a popular destination for those seeking surrogates. She and Nigel were shown profiles of potential donors and surrogates to pick from. "Because the woman carrying the baby is not the biological mother there is less of a chance of her becoming emotionally attached," says Caroline. "We tried to choose donors who had fair skin and looked the most like me, although most of them were dark-skinned so in the end we just chose the prettiest."
The couple's pounds 21,000 "package" allowed them three attempts at impregnation with three different surrogates; nearly pounds 8,000 of this would eventually be paid to the successful surrogate mother. Their first attempt was in April 2013. "I didn't need to be there but flew out with Nigel for moral support," says Caroline. Three embryos, created from harvested donor eggs fertilised with Nigel's sperm, were transferred into the surrogate mother they had chosen. But two weeks after the couple had returned to Britain they received an email to tell them the attempt was unsuccessful.
Devastated, they chose a new egg donor and surrogate and flew out that July to repeat the process. Three embryos were transferred to their new surrogate, with three more frozen to be used later. When their second attempt failed, however, Caroline and Nigel nearly gave up hope. "We agreed to give it one final attempt with a different surrogate mother and the remaining embryos but didn't think it was going to happen," says Caroline. Nigel adds: "We accepted our journey had to stop somewhere."
The embryos were transferred to the third surrogate, Ekaterine, in October 2013. Two weeks later Caroline received an email to tell her that Ekaterine was pregnant. "Nigel and I were euphoric," she says. They were sent monthly reports on their baby's development and when Ekaterine was six months pregnant Nigel and Caroline flew out to meet her.
They learnt that her husband had injured himself and was unable to work, and she was unemployed. "This was her way of earning," says Caroline. "She didn't speak English, which made things difficult. But when we flew out again a couple of months later she was more relaxed."
As part of the clinic's policy, a date was fixed for Ekaterine to have an elective caesarean section at around 37 weeks.
"We flew out the day before and as Grace was delivered I lay down in a room next door and took off my top," says Caroline. "The midwife brought her straight in to me and put her on my chest. It was a wonderful feeling." But afterwards she felt bad about the caesarean section. "Ekaterine was in a lot of pain. She talked about how her milk had come in and she'd had to take drugs to stop it."
Ekaterine had no right to see the baby but Caroline felt she should. "It didn't seem right that she'd gone to all this trouble and couldn't see the baby she had carried," she says. "As I watched her hold Grace tenderly I felt like an impostor, as if I were taking her baby. I had to remind myself Grace wasn't hers because a donor egg had been used."
While Nigel had returned to London earlier, mother and baby flew home in early July, after travel documentation had been completed. The parental order from the Family Court in Britain that makes Caroline Grace's legal mother arrived last month. "It's difficult to stay in touch with Ekaterine because she doesn't have the internet, but we try," she says. "I gave her my gold necklace to thank her and she has a picture of Grace we gave her on her mantelpiece."
Caroline and Grace, meanwhile, have formed an unbreakable bond. "My age does worry me, and sometimes I look at Grace and feel sad that I couldn't carry her myself," says Caroline. "But I'm delighted we finally have a perfect healthy baby."
Surrogacy procedures explained
Egg donation
An egg donor is given daily injections of stimulating hormones to encourage the growth of more eggs for 10 days until she ovulates. During egg collection a needle is introduced through the back of her vaginal wall into her ovaries. The eggs are removed with a gentle suction pump and fertilised with sperm in a laboratory dish through in vitro fertilisation (IVF).
Egg freezing
A woman can freeze her eggs for later use. Eggs are removed in the same way as they are for egg donation. They are dehydrated before freezing and can be stored for up to 10 years in liquid nitrogen at minus 196 degrees centigrade. When the woman wants to have a baby the egg is thawed slowly before being warmed up again and injected with sperm to fertilise.
Traditional surrogacy
The surrogate mother acts as both the egg donor and the surrogate, so is also the biological mother. The biological father gives a sample of sperm that is transferred into the uterus of the surrogate with a process called intrauterine insemination (IUI) in the hope that fertilisation will take place naturally.
Gestational surrogacy
The embryo is created using the biological father's sperm and an egg donor through IVF, meaning the surrogate is not the biological mother. Resulting embryos are transferred into the surrogate mother's uterus with a plastic catheter using a process called uterine embryo transfer (UET) around three days after fertilisation.
