To a food processor or blender, add the pineapple, lemon and lime juices and the sugar; blend until smooth.
Transfer the mixture to a gallon-size storage bag and freeze for 2 hours or until the consistency is of soft-serve ice cream.
Remove the pineapple mixture from the freezer. Stir in the heavy whipping cream and rum.
Freeze for another 1 to 1-½ hours, until firm.
When ready to serve, remove from freezer and blend mixture until smooth. Serve immediately.