According to the news release, France was arrested at 7:30 p.m. Sunday, held overnight and arraigned Monday morning at the Sag Harbor Village Justice Court, where he was released on his own recognizance.
"Mr. France was observed operating a 2017 Lexus northbound on Main Street failing to stop at a duly posted stop sign," the news release said. "Upon traffic stop, it was determined that Mr. France was operating said vehicle in an intoxicated condition.
"Upon search of his person, due to a lawful arrest, Mr. France was in possession of oxycodone pills."
No other information was immediately available. TMZ first reported the arrest of France in the Hamptons.
"We are aware of an incident that occurred last night and are in the process of gathering information. We take this as a serious matter and will issue a statement after we have all of the facts," NASCAR said in a statement.
France, 56, is the grandson of NASCAR founder William H.G. France and son of Bill France Jr.
He has held the position of chairman and CEO of NASCAR since 2003. His sister, Lesa France Kennedy, runs the family's publicly traded track-operating company, International Speedway Corp.
Jim France -- brother of Bill France Jr. -- and Lesa France Kennedy are believed to own the majority stake in NASCAR, according to public documents, but the family does not comment on its ownership structure.
Under NASCAR's substance abuse policy, Brian France could be suspended and required to go through a recovery program, or he could be subjected to drug testing.