Peel and slice the peaches into a large bowl. Add in the black berries. Sprinkle the sugars, cinnamon, corn starch lemon zest and juice and nutmeg over the fruit. Stir to coat.
Pour into an ungreased 9x9 baking dish. Bake this for 10 minutes, then remove from oven.
Serve this warm or hot out of the oven with either ice cream or whipped cream.