The judge said Brown had reached a civil compromise with the other driver and City News Service says the misdemeanor was dropped Thursday. Brown didn't attend the hearing.
Brown claims he traded insurance information with the woman after his Range Rover struck her Mercedes in Toluca Lake in May.
His attorney says he accidentally gave the woman the wrong insurance information. Mark Geragos also says insurance covered the $800 in damage to the Mercedes.
City prosecutor Patrick Hiscocks objected to the dismissal, saying Brown was still on probation at the time of the crash for assaulting then-girlfriend Rihanna in 2009.
He also says the other driver felt she was "castigated" on social media after the crash.