The 37-year-old suffered the injury on 4 June during the first innings of Sussex's County Championship Division Two win over Worcestershire.
It was only the Australian's second Championship appearance of the season after he missed three of Sussex's previous four games with a knee injury.
Magoffin took 5-51 in his only other game as Sussex lost at Nottinghamshire.
Sussex are currently fifth in the Division Two table, with two wins from their opening five matches.
Sussex seamer Steve Magoffin will be out for six to eight weeks after suffering an Achilles tendon injury.