In a small saucepan add the blackberries, sugar, and water. Heat on medium high and mash well. Heat until the mixture is thicker. Strain through a fine mesh strainer and chill.
In a cocktail shaker, add the blackberries, blackberry syrup, limes, and mint leaves. Muddle well, then all the rum and a few ice cubes. Shake well to combined.
Strain the mixture into your glasses, then top with VOVETI prosecco.