The 24-year-old Baysse suffered the injury in a friendly against amateurs Plabennec on Wednesday and will undergo surgery in a couple of weeks, the club said on their website (www.sb29.fr).
"Considering the seriousness of the injury, he probably will be out of action until the end of the season," the club said.
Baysse, who could play as centre back or full back, has started 18 out of the 19 league games this season for Brest, 14th in the league standings on 21 points, five above the relegation zone.
French media have reported that Brest could sign Olympique Marseille's reserve centre back Leyti N'Diaye on loan for the next six months.
Marseille forward Florian Raspentino told French radio he was also on his way to Brest on a six-month loan deal.
(Reporting by Gregory Blachier, editing by Ed Osmond)