Originally taken in the third round by the Rams in the 2015 draft, Brown emerged as a key piece of the offensive line last season.
The 25-year-old started a career high 16 games at right guard in 2017 after playing 20 games combined in 2015-16.
"[Brown's] a young player that has a lot of upside, a lot of ability and we're very encouraged," Rams head coach Sean McVay told Lindsey Thiry of the Los Angeles Times in January.
The Rams were a pleasant surprise last season, winning the NFC West with an 11-5 record for the first time since 2003. They have added talent to their roster this offseason, including Ndamukong Suh, Marcus Peters and Brandin Cooks.
Brown's absence will give Jake Eldrencamp and undrafted rookie free-agent Jeremiah Kolone an opportunity to get on the field for the season's first two games.
The Rams will have Brown back in time for their Week 3 game against the Los Angeles Chargers on Sept. 23.