In a medium saucepan over medium heat, combine the cherries with the water, sugar, and lemon juice, and cook until the cherries are very soft and cooked through, about 10-15 minutes.
Remove from heat and let it cool to room temperature, then refrigerate to chill thoroughly.
Puree the cherries and their liquid and put the mixture through a fine sieve.
Once the cherry mixture is chilled, freeze according to your ice cream maker's instructions.