In a medium saucepan, combine the blackberries and 1/4 cup sugar. Cook over medium heat, until the berries have released half of their juices and the juice has reduced, about 10 minutes.
Scoop out half of the mixture into a bowl. Place a sieve over the same bowl and add the remaining berry mixture to the sieve. Press the mixture through the sieve to extract as much juice and pulp as possible, while removing the seeds. Discard the seeds. Cover the puree and chill.
In a large heavy saucepan, bring the remaining 1/2 cup sugar, heavy cream, milk and salt to a simmer, over medium heat stirring occasionally, remove the pan from the heat.
In a large bowl, whisk the eggs. Temper the eggs by slowly ladling some of the cream mixture into the eggs while whisking the eggs vigorously to prevent them from scrambling. Once you've incorporated about half of the cream mixture into the eggs, pour the custard back into the pan with the rest of the cream mixture, stir to combine. Cook the custard over moderately low heat, stirring constantly, until a thermometer registers 170 degrees (Do not boil).
Pour the custard through a sieve into a clean bowl, to strain out the solids or curds. Let the custard cool. Cover and chill at least 3 hours.
Whisk together the blackberry puree and custard then freeze for several hours OR churn in an ice cream maker according to the manufacturer's directions. (I have a Krups ice cream machine, and it took about 35 minutes.)
If you like soft serve, eat it right away. Otherwise transfer to a container and freeze for a couple of hours to firm up. The ice cream may be quite hard when you first take it out of the freezer, so you may want to let it sit a few minutes to soften before trying to scoop.