Combine 3/4 cup water and sugar in a small saucepan and bring to a boil over medium heat, stirring until sugar dissolves, 3 to 4 minutes. Transfer to a bowl and set ver a larger bowl filled with ice water to chill.
Combine blueberries and raspberries in a food processor and puree. Push through a fine strainer and whisk into sugar syrup. Whisk in lime juice. Pour berry mixture into an ice cube tray and freeze for 2 hours.
Just before serving, place ice cubes in clean work bowl of food processor. Pulse until evenly ground but not yet liquefied, about 10 2-second pulses. Spoon into 6 dessert bowls and serve immediately.