In a large saucepan combine the cider, the cinnamon stick, the sugar, and a pinch of salt and boil the mixture for 5 to 10 minutes, or until the liquid is reduced to about 4 cups.
Stir in the lemon juice, strain the mixture through a fine sieve into a bowl, and chill it, covered, until it is cold.
Freeze the mixture in an ice-cream freezer according to the manufacturer's instructions.