Catalyzes the synthesis of cardiolipin (CL) (diphosphatidylglycerol) by specifically transferring a phosphatidyl group from CDP-diacylglycerol to phosphatidylglycerol (PG). CL is a key phospholipid in mitochondrial membranes and plays important roles in maintaining the functional integrity and dynamics of mitochondria under both optimal and stress conditions. 1,2-diacyl-sn-glycero-3-phospho-(1'-sn-glycerol) + a CDP-1,2-diacyl-sn-glycerol = a cardiolipin + CMP + H(+) Belongs to the CDP-alcohol phosphatidyltransferase class-I family.