Halve the melon and remove the seeds. Cut into quarters or eighths, peel, and cut into large pieces.
Place the melon pieces in a large, clean glass jar with a lid. Add the rock sugar, cloves, and lime peel. Pour the grappa over all ingredients, seal the jar, and let it infuse for 4 to 6 weeks, shaking the jar occasionally.
Once the sugar has dissolved, strain the liqueur through a fine sieve into a clean bottle. Rest for at least 6 more weeks so the flavors fully develop. Filter through cheesecloth again if necessary. Add a little sugar solution to taste if you prefer it sweeter.