Combine the pineapple, pineapple juice, and 1/4 cup lime juice in a medium saucepan and cook over low heat until most of the liquid has reduced and the pineapple is very tender, about 20 minutes. Be careful not to let the pineapple burn towards the end of the cooking time. Remove from the heat and puree until very smooth. Chill thoroughly before proceeding. You should have about 1 cup of puree.