Combine water and sugar in a small saucepan, bring to a boil and cook over medium heat until the sugar has dissolved, about 5 minutes. Remove from fire and cool.
Immerse the peaches in boiling water for 1 minute, then pour over cold water and remove the skin. Cut in half, remove the pit. Grind the peaches with a blender into a smooth puree, add sugar syrup, lemon juice and rum, mix.
Pour the mixture into a container and refrigerate for at least 3 hours. Every 30 minutes. Take the container out of the refrigerator and stir the mixture with a whisk.