Six people were said to be seriously hurt, including a 13-year-old girl who was in a critical condition in hospital in Reims, they said. Most of those on board were children.
The driver, who is believed to have fallen asleep at the wheel when the coach crashed in the early hours of Sunday, has been detained, a local prosecutor said.
The British Solus travel coach was on a school trip and carrying 29 children from a school in Alvechurch, just south of Birmingham, and 18 adults in addition to two drivers, the Foreign Office in London said.
Foreign Office Minister Alistair Burt said: “Our thoughts and sincere condolences are with everyone involved in or affected by the tragic coach crash in Northern France earlier today.
“The French authorities are investigating the accident and we are working closely with them.”
Burt said the British ambassador to France, Sir Peter Ricketts, had “seen many of the affected passengers this afternoon and has assured them that we are doing all we can to help them recover and get home.
The deputy ambassador would accompany some Britons back home Sunday, he added, and they would continue to assist those who were on the coach and their loved ones back in Britain.
Burt also thanked French emergency services “for all they have done”.
The family of 59-year-old teacher Peter Rippington said they were “devastated at the tragic loss of Peter, a wonderful husband, father, son, brother, son-in-law, brother-in-law, uncle, friend and teacher”.
They were still concerned about the health of his wife Sharon, who was also injured they added. Their daughter Army had escaped with minor injuries.
The Interski Snowsport school which organised the trip said it was “saddened and distressed” by the accident.
It said the coach “veered from the motorway before coming to rest on its side at the foot of an embankment”.
The injured were taken to hospitals in Reims and Chalons.
The coach was the only vehicle involved in the crash, which happened at around 3am (0200 GMT) on a motorway near Chalons-en-Champagne.
The road was partly closed for about four hours as 100 firefighters, 20 police and four emergency teams helped the victims.
Some 25 passengers are being treated for shock, the prosecutor said, adding that as many as possible would be repatriated on Sunday.
They were taken to the small nearby town of Fagnieres, but a rescue vehicle was expected in the crash area late Sunday afternoon to bring home those judged fit to travel.
The prosecutor said the driver, who was slightly injured, had tested negative for drink or drugs.
Local officials said a second British coach that came upon the accident had taken the victims on board to shelter them from the rain.