Preheat the oven to 375?F.
In a large bowl, combine the peach slices with 1 cup of the sugar, the peach liqueur, and the rum. Stir to mix. Cover and refrigerate for at least 1 hour. (This can be done a day ahead.)
Using an electric mixer fitted with a wire whip, beat the egg whites on medium speed with the cream of tartar and the salt until soft peaks form, about 2 minutes. Add the remaining cup sugar and the vanilla and beat until stiff peaks form, about 1 minute. Add the flour and the cinnamon and beat for 2 minutes, scraping down the sides of the bowl as needed. Pour the batter into an ungreased angel food cake pan. Bake until golden brown and springy when touched, 30 to 35 minutes. (The top of the cake may crack a little.) Remove the pan from the oven and turn it upside down over the neck of a bottle. Let cool completely.
Remove the cake from the pan and slice into 12 equal pieces. To serve, place the slices of cake on dessert plates, spoon some of the peaches over, and garnish with whipped cream and a sprig of mint.
Yield: 1 cake to serve 12