In a saucepan, combine the cranberries, orange juice, and ½ cup of the water. Cook over medium heat until cranberries pop open, 5 to 7 minutes. Coarsely mash the cranberries, still in the saucepan, with the back of a wooden spoon. (You should have about 1½ cups puree.)

Add the sugar, lemon juice, orange zest, and remaining water. Bring to a boil over medium-high heat.

Reduce heat to low and simmer, stirring often, until a syrup forms, about 15 minutes.

Let cool and refrigerate for at least 6 hours, or as long as overnight.

Freeze in an ice-cream maker according to the manufacturer’s instructions.