Soup: In a saucepan, bring water, sugar, blackberries and lemon juice to a boil over medium-high heat. Simmer for approximately 10 minutes, then add mint leaves. Remove from heat and let cool. Blend mixture together in a mixer or food processor, then strain through a sieve.
Sorbet: In a medium saucepan, bring sugar, water and corn syrup to a boil over medium-high heat. Let cool, then stir in lime juice and sour cream. Churn in an ice cream maker until proper consistency is achieved.
To serve: Divide blackberry soup into four equal parts and pour into separate bowls. Put 1 scoop of sorbet in the middle of each bowl and drizzle with champagne.