Meester was scheduled to have surgery Wednesday.
Meester, who has started 108 games for Jacksonville in seven years, injured his ankle during the team's first full-pads practice Monday night. The Jags initially thought it was just a sprain, but tests later determined a more severe injury.
"It's tough for Brad because he loves to play the game and loves to play with us," guard Vince Manuwai said Tuesday. "But the good part is it's only eight weeks, and he could be back during the season. It's not a total loss. It's just a loss for now."
Dennis Norman, a versatile backup who started six games the last two years, will move into Meester's starting spot. Norman started the last five games at center in 2005, filling in for Meester after he tore a biceps tendon.
Norman started at left tackle last season when Khalif Barnes was benched following his arrest on drunken driving charges. Norman played well during both stints.
"It doesn't feel different with Dennis in there," Manuwai said. "Dennis has been in there with us in the games. He started games and played in the playoffs. It's not like he played sparingly. He played tackle, guard, center. He pretty much knows every aspect of the offense on the line.
"For him, it's probably good that now he gets to focus on one position."
Meester, a second-round draft pick in 2000 from Northern Iowa, started the first 92 games in his NFL career. He played guard for three seasons, then moved to center in 2003.
He has anchored one of the better lines in the league. The Jags rank sixth in the NFL in rushing between 2003 and 2006, averaging 131.6 yards. Jacksonville, led by running backs Fred Taylor and Maurice Jones-Drew, set a franchise record with 2,541 yards rushing last season.
Although quarterback Byron Leftwich missed 10 games last season with an ankle injury, the Jags were relatively healthy on offense. The defense, however, lost three starters - linebacker Mike Peterson, defensive end Reggie Hayward and safety Donovin Darius - and several reserves.
Coach Jack Del Rio had hoped the team would be able to stay healthy during training camp and the preseason.
"We're back at it again," Jones-Drew said.
Also Tuesday, several players missed practice because of injury.
Hayward (Achilles' tendon), running back Greg Jones (knee), receiver Charles Sharon (head) and defensive linemen Marcus Stroud (ankle), Bobby McCray (quadriceps), Paul Spicer (back) and Jeremy Mincey (calf) sat out the morning session. Cornerback Scott Starks has a broken bone in his left hand, but practiced with a cast.