Jones, 25, is to be sentenced in mid-September before Judge Bloom, Forbes said.
The maximum sentence on the misdemeanor charge of simple possession of cocaine is six months, he said. Jones, who has already served three months in jail, would get credit for that time, he said.
Jones has a history of issues with drugs and alcohol.
In 2008, he was arrested after crashing his car while driving under the influence. In 2011, he was arrested on charges of misdemeanor possession of heroin.
When he was arrested in March on drug charges along with a Detroit man, the mayor issued a statement saying that he was heartbroken and relieved that his son had been arrested because it could save his life. He cited Zac's struggle with addiction and the heartache it had caused those who love him.
Zac and Moldavian Harris, 24, of Detroit, were arrested during a routine traffic stop on Interstate 77 north of Charleston when officers stopped a speeding Mazda 3. A search of the vehicle turned up 25.7 grams of cocaine.
Contact writer Charlotte Ferrell Smith at charlo...@dailymail.com or 304-348-1246.