TMZ Sports broke the story ... France was busted in the Hamptons on Sunday evening. Cops say he was drunk and illegally in possession of prescription oxycodone pills.

France -- widely considered one of the most powerful people in pro sports -- has issued a statement apologizing.

"I apologize to our fans, our industry and my family for the impact of my actions [Sunday] night. Effective immediately, I will be taking an indefinite leave of absence from my position to focus on my personal affairs."

NASCAR says Brian's uncle, Jim France, has "assumed the role of interim chairman and chief executive officer."

Story developing ...