Combine first 4 ingredients in a medium saucepan; bring to a boil. Reduce heat; simmer for 5 minutes. Cool, and discard mint.
Place half of peach mixture in a blender; process until smooth. Pour pureed peach mixture into a 13 x 9–inch baking dish. Combine remaining peach mixture, juice, and rum in blender; process until smooth. Add to baking dish.
Freeze 8 hours or until firm. Remove mixture from freezer; scrape entire mixture with a fork until fluffy. Serve immediately. Garnish with peach slices and mint sprigs, if desired.
Note: Nutritional analysis is less than 1% from fat.
My version using nectarines instead. I used half the sugar because I don't like drinks too sweet. Only a half a cup of lime. Blended all ingredients, including the booze and mint leaves. Then froze it over night. You couldn't even see the mint leaves, except some specs. It was wonderful!