I used a home-brewed cold-brew coffee concentrate, but if you don't have the means to make your own, it's also available commercially. This isn't the cold-brewed stuff that you drink as-is. This is the concentrated stuff that you add to hot or cold water to make your coffee. A tablespoon or two is enough to make a full cup of coffee.
If you can't find a coffee concentrate, then use espresso or a concentrated 1/4 cup of instant coffee. Or, just leave it out and make chocolate ice cream without the coffee. Add 1/4 extra cup of milk, if you do that.
I bought a Filtron coffee maker a while back, and that's what I used to make the cold-brew concentrate. I remember my parents having an old model made from glass, and I loved watching the coffee drip-drip-drip slowly through the grounds. The one I bought is made from plastic and it's not quite the same as I remember, but it still makes a coffee concentrate that's smoother than a hot-brewed cup.
Black Cocoa Ice Cream
1 3/4 cups heavy cream, divided
1 cup milk
3/4 cup sugar
1/2 teaspoon salt
1/4 cup black cocoa
1/4 cup cold-brew coffee concentrate
2 tablespoons cornstarch
1 tablespoon vanilla extract
Combine 1 cup of heavy cream, along with the milk, sugar, salt, and cocoa in a heavy-bottomed saucepan, stirring as needed, until it reaches a simmer.
Meanwhile, mix the cold-brew coffee concentrate and the cornstarch.
When the cream mixture has come to a simmer, give the coffee-cornstarch mixture a little stir, just to make sure it's well blended.
Whisk the coffee mixture into the cream mixture, and continue heating until it returns to a simmer and you can feel it thickening.
Turn off the heat and add the additional 3/4 cup heavy cream and the vanilla. Stir well. Let the mixture cool, then refrigerate until well chilled.
Churn in your ice cream maker according to the manufacture's instructions.