Rinse kumquats, rub dry, dice finely, removing seeds. Place kumquats and preserving sugar in a large enough bowl and mix well. Cover and set aside for 3 to 4 hours.
Place kumquats and jelling sugar in a large saucepan. Add rum, lime juice and vanilla sugar; bring to a boil, stirring 5 minutes until bubbling.
Pour jam into screw-top jars, turn to the other side, set.
ROUND. Ripe fruit aind uniformly orange-red.