The goalkeeper missed tonight's 1-0 victory over Coventry with the issue and Carlos Corberan confirmed Palmer will be out of action until mid-March.
Palmer felt pain during Albion's warm-up with goalkeeper coach Gary Walsh. The club sent the academy graduate for tests on Friday morning and results confirmed a problem.
"Six weeks unfortunately," Corberan said after the 1-0 win courtesy of Grady Diangana's goal. "Yesterday while he was warming up with the goalkeeper coach, he deflected one ball with the foot and he felt something in the knee and the ankle.
"Today, we had a test in the morning to see how he'd react from his pain.
"He was failing the test, he kept failing the test and he started to get inflammation. So we sent him to have a scan, and the scan said he would be out of the team for six to seven weeks."
Palmer has enjoyed inspired form under Corberan and been a big part in the team's turnaround in recent months.
He was replaced by David Button, who Palmer came in for in October, and Button achieved a deserved clean sheet with a faultless display against the Sky Blues.
Corberan added: "Alex for me has helped the team to achieve points, but when you lose a player the key to have the next one ready to help the team.
"Today I am very pleased with David because it has not been an easy season for him. He gave the step that he had to give.
"He is an important member of the dressing room. They have one level of commitment - they are suffering when they are watching their teammates suffer. They enjoy when their teammates succeed. They know that he was suffering and today they were very pleased for him - individually but also pleased because he was helping the team."