The British and Irish Lions star picked up the injury in the first half of the Ospreys defeat to London Irish. Scans have confirmed ligament damage which requires a lay-off of eight weeks.
Jones now joins fellow front rower Gethin Jenkins on the sidelines after the Blues prop had toe surgery.
“The scans have confirmed our suspicion of significant ligament damage and soft tissue trauma to the left elbow,” Ospreys physio Chris Towers said.
“We’ve discussed the case at length, both with orthopaedic specialists and the WRU medical team, and expect that Adam is likely to be out for around eight weeks.”
With Wales kicking off their RBS 6 Nations campaign against England at the Millennium Stadium on Friday 4 February, Jones joins a growing list of casualties with midfielders Tom Shanklin and Andrew Bishop also suffering from knocks.