Begin by preheating your oven to 350° and preparing a standard Bundt pan with non-stick cooking spray, then dust it lightly with flour and shake out any excess.
In a large bowl, use an electric mixer to combine 1 (18 1/4 ounce) box French vanilla cake mix, 1 pint melted coffee ice cream, and 3 eggs. Beat these ingredients on medium speed for 3 minutes until well combined and smooth.
Pour the batter into your prepared pan and bake at 350° for 40 minutes, or until a toothpick inserted in the center comes out clean. Once baked, allow the cake to cool in the pan on a wire rack for 20 minutes before removing.
While the cake cools, prepare the frosting by beating together 1 (8 ounce) package softened cream cheese, 1/4 cup sour cream, 3 tablespoons softened unsalted butter, 2 tablespoons coffee liqueur, and 1 teaspoon vanilla extract in a large bowl on medium speed.
Continue beating for about 2 minutes until the mixture becomes light and creamy. Gradually add 1 1/2 cups confectioners’ sugar on low speed, mixing for 1 to 2 minutes until fully incorporated. Refrigerate the frosting while the cake finishes cooling completely.
Once the cake has cooled to room temperature, spread the frosting generously over the top and sides of the cake.
Return the frosted cake to the refrigerator and keep it chilled until ready to serve. Just before serving, dust the entire cake with 1 tablespoon of cocoa powder using a fine sieve or sifter for an elegant presentation.