Peel the papaya and cut in half. Spoon out the seeds and keep them in a small container in the refrigerator.
Cut the papaya into chunks and reduce to a liquid puree in the food processor. Add the sugar and lime juice. If the puree is too thick, stir in a little water, but be very careful that you do not add too much water or you will lose the consistency.
Serve in fruit cups with a dab of the reserved seeds in the center of each serving.