Bardet (AG2R-La Mondiale) started the stage in fourth place overall and only 30 seconds behind Primož Roglič (Jumbo-Visma).
However, the 29-year-old crashed on a descent, with Nairo Quintana (Arkéa-Samsic), and while he quickly got back on his bike, Bardet had to sit down due to a bout of dizziness.
It meant the 2016 Tour de France runner-up dropped to 11th overall - three minutes behind Roglič - and the AG2R La Mondiale team later announced the Frenchman is out of the race, due to the concussion he suffered in the crash.
"Romain suffers from a concussion, the intensity and signs of which he presented after the stage required our medical team to have him undergo a brain scan at the Clermont Ferrand University Hospital," AG2R team doctor Eric Bouvat said.
"This examination did not reveal any lesions. However, it is necessary for Romain Bardet to stop his sports activities and he will not be able to start the stage tomorrow.
"His date of resuming the competition will be defined according to the progress [of his injury]."
The 2020 Tour de France continues with Stage 14, a 194-kilometre day in the mountains of the Massif Central, from Clermont-Ferrand to Lyon. Watch the race on the SBS, SBS On Demand and the SBS ŠKODA Tour Tracker - with coverage starting from 8:30pm (AEST).
