Hodges said Friday that the sides recently finalized the deal.
Hodges played in eight games last season for Cleveland after Dave Zastudil was injured and averaged 39.8 yards on 45 punts. He assumed the starter's job in training camp this year when Zastudil couldn't kick after undergoing offseason knee surgery. The Browns recently waived Zastudil.
Hodges is averaging 44 yards on 51 punts and leads the NFL with 11 punts inside the opponent's 10-yard line.
Hodges also made one of the Browns' biggest plays, running 68 yards on a fake in an upset of the New Orleans Saints.