Brown did not say when the incident happened, but the Orange County Register, citing a public records search, reported that Brown was arrested in the Louisville area in February 2017.
Brown pleaded guilty to speeding and aggravated DUI, first offense, and was fined and sentenced to four days' house arrest, according to the Register. Charges of careless driving, marijuana possession and drug-paraphernalia possession were dropped.
"It was kind of like a careless mistake," Brown told reporters, according to the Register. "But it happened, and all I can do now is be ready for when it's time for me to come back."
He said he was surprised when the NFL suspended him for the first two games of the 2018 season based on "intel" he had been receiving from the NFL Players Association.
Brown practiced with the starters at right guard on Thursday, and coach Sean McVay said he anticipated that Brown would reclaim his spot after he served his suspension.
Brown, a fourth-year pro, can return to the active roster on Sept. 17 in preparation for the Rams' Week 3 game against the Los Angeles Chargers.
Last season, Brown started 16 games as the Rams became the only team in the NFL to start the same five linemen for 15 consecutive weeks. The group allowed 28 sacks and helped spring running back Todd Gurley for 1,305 yards rushing and 13 touchdowns.
ESPN's Lindsey Thiry contributed to this report.