The school announced that Barrett was cited with a misdemeanor offense of OVI and will serve the suspension for top-ranked Ohio State's game on Nov. 7 against Minnesota. The defending national champion Buckeyes do not have a game this weekend.
Barrett, 20, was arrested after trying to avoid a DUI checkpoint near the Ohio State campus, a source told ESPN. He failed a Breathalyzer test and was released to teammate and fellow quarterback Cardale Jones, according to the source.
The arrest was first reported Saturday by ElevenWarriors.com.
Buckeyes coach Urban Meyer decided on Barrett's punishment, Ohio State athletic director Gene Smith told ESPN. Smith also said Barrett will be eligible to return for Ohio State's road game at Illinois on Nov. 14 and its Nov. 21 showdown against No. 6 Michigan State at the Horseshoe.
Barrett, a sophomore from Wichita Falls, Texas, had regained the starting job from Jones before last week's 49-7 victory over Rutgers. He took over the offense against Rutgers after Jones started the first seven games of the 2015 season. Barrett completed 14 of 18 passes for 223 yards with three touchdowns in what might have been Ohio State's most impressive victory of the season.
Barrett was the starter in 2014 before breaking his ankle in the final regular-season game against rival Michigan.
Information from ESPN's Brett McMurphy was used in this report.