As a child, I remember visiting the farm of a family friend and having these pickles. They tasted just like candy! This recipe comes from "The Joy of Pickling" (an excellent resource for canners) and tastes just like the ones I remember. It's similar to watermelon rind pickles in its prep and ingredients. Don't take any shortcuts on the rinsing! Pickling Lime can sometimes be found in the canning section of WalMart or you can order it online.
In a large bowl, stir the lime into 3 cups water. Add the melon balls and let the bowl stand at room temperature for 4 - 5 hours.
Drain the melon balls, rinse them well, and cover them with fresh water. Let the bowl stand for 1 hour, then drain them again. Give the melon two more hour long soaks in fresh water. Drain melon well.
In a nonreactive pot, combine the vinegar, 1 cup water, sugar and dry spices and ginger tied in cheesecloth. Bring the syrup to a boil, stirring to dissolve the sugar and add the melon balls. Bring the syrup to a boil again, reduce the heat, and cover the pot. Simmer the melon about 1 hour until the balls are translucent around the edges.
With a slotted spoon, divide the melon balls between two pint mason jars. Boil the syrup until it thickens a little and remove the spice bag. Pour the hot syrup over the melon balls, leaving 1/2 inch headspace. Close the jars and process for 10 minutes in boiling water bath.