Sarrazin was airlifted to hospital after losing control during a practice run on Friday, bouncing off the surface and sliding a long way. He underwent surgery the same day to reduce an intracranial haematoma.
"His state of health is stable. He was discharged from intensive care and admitted to a care unit," the FFS posted on X.
"The rest of the medical assessment (foot and knee pain) is very reassuring and, despite the violence of the impact, no other injuries have been noted."
The Frenchman, who faces a long period of recovery, could be repatriated to France on Friday, the statement added.
"The length of his unavailability is currently undetermined," the FFS said.