* Doctors operate to remove hematoma from brain
* Schumacher condition improved but still fragile
* Seven-times champion is most successful F1 racer ever (Adds further quotes and background)
By Lucien Libert
GRENOBLE, France, Dec 31 (Reuters) - French doctors treating Michael Schumacher for brain injuries sustained in a ski fall said the seven-times Formula One world champion was in slightly better condition on Tuesday after an overnight operation, but that he remained fragile.
The 44-year-old German is battling for his life after slamming his head against a rock while skiing off-piste in the French resort of Meribel on Sunday, an accident which triggered an outpouring of concern among fans around the world.
Doctors treating him at a hospital in the eastern city of Grenoble said his condition had stabilised enough by late Monday to carry out a new operation to treat the effects of internal bleeding within Schumacher's skull.
"The situation is more under control than yesterday but we cannot say he is out of danger," Jean-Francois Payen, head anaesthetician, told a news conference at the CHU hospital in the eastern French city of Grenoble.
"We have won some time but we must continue an hour-by-hour surveillance... It is premature to speculate on his condition," he said, adding that Schumacher was still in a critical state and suffering from severe lesions and contusions.
Emmanuel Gay, head of the hospital's neurosurgery service, said the operation carried out around 10.00 pm (2100 GMT) on Monday had successfully removed a large hematoma - the medical term for a build-up of blood - from his brain.
"It was larger and more accessible (than others) ... We judged we could remove it without taking any risks," Gay said.
He said the operation was designed to reduce, within Schumacher's skull, the pressure on the brain.
Doctors said the fact that the retired motor racing champion was wearing a helmet at the time of the accident had at least enabled him to make it to the hospital alive.
Payen said the medical team in Grenoble had discussed the operation with Schumacher's family. He added that the condition of the motor racing great was still too fragile to consider transferring him to another hospital for the time being.
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Schumacher is under the care of Professor Gerard Saillant, a brain and spinal injury expert who is also president of the International Automobile Federation (FIA) Institute.
Saillant said it was still impossible to say how Schumacher's condition would progress in coming days.
"We are a little less worried than yesterday but I'm sure you understand that the situation could change this evening or tomorrow," he told the news conference.
Schumacher, who lives in Switzerland with his wife and two children, is the most successful Formula One driver of all times with a record 91 race victories in a career spanning more than two decades.
Schumacher left the sport last year after a less successful three-year comeback with Mercedes following an earlier retirement from Ferrari at the end of 2006.
French authorities have opened an investigation into the accident, which took place as Schumacher was out skiing with his teenage son.
Ferrari always used to have an annual January gathering with their drivers in the Dolomites and Schumacher, a fitness fanatic, impressed with his skiing ability.
German Chancellor Angela Merkel had been deeply shocked to learn of his accident, her spokesman said on Monday as expressions of concern poured in from fans, former team-mates and rivals.
Former British Formula One driver David Coulthard said he believed Schumacher had not won the full recognition he merited for taking his sport to new heights.
"I only hope Michael Schumacher pulls through so that he can see all the nice things people are saying about him," he wrote in the Daily Telegraph newspaper. (Additional reporting by Mark John and Natalie Huet in Paris and Alan Baldwin in London; Writing by Mark John and editing by Natalie Huet and Ed Osmond)