Line strainer with two layers of dampened cheesecloth. Set over large bowl. Combine berries * and water in medium saucepan. Slowly bring to boil. Reduce heat and simmer gently until soft, about 10 minutes. Pour into cheesecloth-lined strainer. Let stand until juice has drained into bowl, about 30 minutes. Gently squeeze pulp to extract remaining juice. Measure berry juice into heavy medium saucepan. Add 1 cup sugar for each cup juice. Cook over low heat, swirling pan occasionally, until sugar dissolves. Increase heat and boil syrup 2 minutes. Cool completely. Mix 2 1/2 cups syrup with cream. Refrigerate until well chilled. Process blackberry mixture in ice cream maker according to manufacturer's instructions; ice cream will be soft. Freeze in covered container several hours to mellow flavors. Garnish with fresh berries and mint leaves.
* Can substitute unsweetened frozen blackberries, thawed.