In a small saucepan, combine the sugar and 3/4 cup of water. Cook over moderate heat, stirring constantly to dissolve the sugar, and then bring just to a boil. Remove from the heat and let cool to room temperature. Pour the syrup into a bowl set over a larger bowl of ice and stir until cool. Add the limejuice and puréed papaya. Scrape the mixture into an ice cream maker and freeze according to the manufacturer's directions.