Peel the papaya and cut in half. Spoon the seeds out and discard the seeds and peels.

Shred the papaya flesh using a Y-shaped vegetable peeler or cut into thin pieces using a paring knife. The pieces should be about 1 inch wide and a couple of inches long. Or use a box grater to grate the flesh if you'd like.

Peel the limes and reserve the flesh of 1 lime for juicing. Tie the peels up in cheese cloth and set aside. Alternatively, you can zest the limes using a coarse grater, that way you wont need the cheese-cloth.

Add the shredded papaya, three cups of sugar, rum and water to a 6 quart stock pot. Bring the pot to a boil over high flame. Once the mixture starts boiling, cover the pot and turn the burner down low. Let simmer for 1 hour.

While the papaya simmers, heat the remaining 1 cup of sugar in a small sauce pan over medium/low heat. Stir the sugar occasionally and watch it closely. You'll start to notice an amber pool of liquid where the sugar starts to melt. Keep stirring. The sugar will start to clumps. Eventually more liquid will form and those clumps will turn to liquid. Keep stirring until all of the sugar has dissolved and the liquid is deep-amber in color. Proceed quickly to the next step so the sugar doesn't harden!

Uncover the pot with the jam and quickly use a pair of tongs or large spoon to remove the lime peels/cheesecloth from the pot (if you used lime zest, don't worry about this step!). Pour the caramelized sugar into the pot. The sugar may sputter and harden when it's added but it will become liquid again. Just keep stirring!

Stir in the molasses and cover the pot again. Let the papaya jam cook for another 20-30 minutes, or until most of the liquid has evaporated. The papaya will reduce in volume and be coated in amber-colored syrup.

Turn the heat off from under the pot and let the jam cool for 20-30 minutes before handling. Transfer the jam into glass canning jars and let them cool to room temperature. Once they are cool, cover each jar with a lid and store the papaya jam in the refrigerator.