Vicky and Andy Lamb had been keeping a constant vigil in the neonatal intensive care unit of the John Radcliffe Hospital in Oxford, since their birth on Friday May 14.
But this evening their spokesman, Max Clifford, gave the sad news that one of the two boys had passed away.
He said: "Andy and Vicky are so upset about losing one of the little boys and are hoping and praying the others will be okay."
The Lambs had conceived four girls and two boys with the help of fertility treatment.
The children were born 14 weeks early. The smallest child weighed just 1lb 5oz while the largest was only 1lb 15oz – the average baby weighs 7lb 5oz at birth.
The Lambs, who are both 31 and run their own company specialising in health and safety, have a five-year-old daughter, Gracie, who they also conceived using the fertility drug Clomid.
They have already named all six babies. Only five of the last sextuplets born in Britain – to the Conway family in Northern Ireland last May – survived beyond sixty days.
Mr Clifford had said earlier on Friday: "Although Andy and Vicky have a little girl, they were absolutely thrilled that this has happened and they are just hoping and praying that everything is going to be OK but they realise that they are praying for a small miracle.
"So far everything's going the way that they hoped and prayed it would but we have to wait and see. Everything at the moment is down to the surgeons, doctors, nurses and specialists who are doing such a wonderful job for them.
"They are being well looked after but it's a very very critical time."
Asked when the couple might be able to take their babies home, Mr Clifford said: "It is a long way off. They are fighting for their lives."
The couple's daughter Gracie is being cared for by neighbours while they stay in hospital.
Most of those living close to the family's modern terraced house in Abingdon, Oxfordshire, said they had watched Mrs Lamb's bump grow but had no idea she was expecting sextuplets.
One neighbour, who asked not to be named, told of the couple's excitement as they prepared for the birth.
"They didn't seem nervous at all. They were excited before they went in, like you are when you're having kids," he said.
"Every time I saw her she was getting bigger and bigger, and I guessed it must be twins or triplets, but I never dreamed it could be sextuplets. You just don't expect it to happen."
Sextuplets occur in just one in every 4.5 million pregnancies and the births of the Lambs' has been meticulously planned over several weeks.
Dr Kenny McCormick, a Consultant Neonatologist at John Radcliffe Hospital said the next few days and weeks will be critical.
"Babies that are born this early need a lot of specialist care," he said. "They are receiving round the clock intensive care and specialist nursing at the moment and their condition is constantly monitored."
While medical advances have made it possible for a much higher number of premature babies to survive, only three sets of sextuplets have survived in the UK in the past 27 years.
The first to do so were the Walton sisters, who also became the world's first surviving all-girl sextuplets. Hannah, Lucy, Ruth, Sarah, Kate and Jenny were conceived following fertility treatment and born in Liverpool to Graham and Janet Walton in 1983.
The Walton sextuplets were followed by the Coleman sextuplets in 1986 and the Vince sextuplets in Leeds in 1993.
Last year in the US, Nadya Suleman gave birth to octuplets in California. Later it emerged that she was a single mother who already had six children, all conceived through IVF.
Sextuplets were born in May last year in Belfast to Austin and Nuala Conway, of Dunamore, County Tyrone, who called the babies "beautiful little miracles".
The four baby boys and two girls, conceived naturally without the aid of IVF, were born in the space of five minutes.
Sixty days later, Kerrie Mae Conway, the youngest of the six, died in hospital. They too were born 14 weeks premature and weighed from 1lb 7oz to 2lb 2oz.