In a small saucepan, cover the fruit with cold water and bring to a boil. Drain. Cover the fruit with cold water and bring to a boil again. Drain and set aside.
In the same saucepan, combine 1 cup water and the sugar, bring to a boil, reduce heat and simmer for 10 minutes.
Pierce each piece of fruit 2 or 3 times with a paring knife. Drop the fruit into the sugar syrup and continue to simmer for 15 minutes for kumquats or 20 minutes for lemons.
Remove from heat and leave the fruit steeping in the syrup unrefrigerated for 8 hours or overnight.
Bring the syrup and fruit to a boil, then reduce to a simmer for 10 minutes. Cool and store in a glass jar. Fruit and syrup will keep in the refrigerator for 3 months.