1. Combine 3 cups blackberries with 1 cup sugar and the water in a medium nonreactive saucepan. Bring to a boil over medium heat. Reduce heat to a simmer, and cook, stirring constantly, until sugar dissolves and berries begin to fall apart, about 4 minutes.

2. Remove from heat. Pass berry mixture through a fine sieve into a medium bowl, gently pressing down on the solids with a wooden spoon to release as much liquid as possible. Measure out 1/2 cup solids, and set aside; discard remaining solids. Measure out 1 2/3 cups strained puree, and return it to saucepan; reserve any remaining puree for serving. (Store reserved puree in an airtight container in the refrigerator until ready to use.)

3. Place vanilla bean and scrapings in a saucepan, and add milk. Bring to a gentle boil over medium heat, and remove from heat. Discard vanilla pod.

4. Prepare an ice bath, and set aside. Combine egg yolks and the remaining 1/3 cup sugar in the bowl of an electric mixer fitted with the whisk attachment. Beat on medium-high speed until mixture is pale yellow, 3 to 5 minutes.

5. Using a measuring cup or ladle, slowly pour about 1/2 cup hot milk mixture into the egg-yolk mixture, beating constantly on low speed until blended. Continue adding milk mixture, about 1/2 cup at a time, beating until thoroughly combined after each addition.

6. Return mixture to saucepan. Cook over medium heat, stirring constantly with a wooden spoon until thick enough to coat the back of the spoon and hold a line when drawn across the back of the spoon with your finger, 6 to 8 minutes.

7. Remove from heat; add reserved strained puree, and immediately stir in the cream to stop the cooking. Pour through a fine sieve into a medium bowl set in the ice bath; let cool completely, stirring occasionally. Cover bowl with plastic wrap, and place in the refrigerator until thoroughly chilled, at least 30 minutes or overnight.

8. Freeze mixture in an ice-cream maker according to manufacturer's instructions; it should still be slightly soft. Stir in the reserved 1/2 cup blackberry solids and remaining 1 cup blackberries; spin a few times in ice-cream maker to distribute evenly. Transfer mixture to an airtight container, and place in the freezer until ice cream is completely set, at least 4 hours and up to 1 week.