Melt the chopped chocolate in a double boiler or in the microwave until smooth.
In a medium-sized saucepan, whisk together the milk, heavy cream, and 1/2 cup of sugar. Heat the mixture over medium heat until it begins to steam.
In a mixing bowl, whisk together the egg yolks and remaining 1/4 cup of sugar until pale and thick.
Slowly pour the hot milk mixture into the egg mixture, whisking constantly.
Return the custard mixture to the saucepan and cook over medium heat, stirring constantly, until the mixture thickens and coats the back of a spoon.
Strain the custard through a fine-mesh sieve into a clean mixing bowl. Stir in the melted chocolate and freshly brewed coffee.
Cool the custard to room temperature, then cover and refrigerate until chilled.
Churn the custard in an ice cream maker according to the manufacturer's instructions.
Transfer the churned ice cream to a container and freeze until firm, about 2-3 hours.