The six - four girls and two boys - were born last Friday to Andy and Vicky Lamb, both 31, from Oxford.
One of the boys died this morning, Mr Clifford said.
The babies' parents were said earlier today to have been hoping for a "small miracle" as the sextuplets fought for their lives.
Mrs Lamb gave birth shortly after 11pm last Friday. She had fallen pregnant after taking the same fertility drug she used before first child Gracie, now five, was conceived.
She and her husband Andy have been anxiously waiting for good news as the tiny babies, whose birth weights ranged from 1lb 5oz (590g) to 1lb 15oz (885g), struggled to survive.
Mr Clifford said earlier today: "They are just incredibly grateful for the John Radcliffe Hospital, the doctors and the nurses."
Talking about when the couple might be able to take the babies home, Mr Clifford said: "It is a long way off. They are fighting for their lives, they are still critical."
In an interview with GMTV, Mr Clifford added that Mr and Mrs Lamb were "hoping and praying that everything is going to be okay - but they realise they are praying for a small miracle".
Live births of sextuplets are rare, occurring approximately once in every 4.5 million pregnancies.
According to the Multiple Births Foundation charity, the last surviving set of sextuplets in Britain were born in 1993.
The latest set were delivered by a large team of doctors and nurses following several weeks of planning.
Kenny McCormick, consultant neonatologist at the hospital, said: "The next few days and weeks will be critical. Babies that are born this early need a lot of specialist care.