One hour before starting your cake, turn the freezer to its highest setting.
Heat oven to 350° F. Grease and flour two 8-inch round cake pans. In a large mixing bowl, combine the cake mix, eggs, and oil, along with the amount of water called for by the package directions. Beat until well combined. Divide the batter between the pans. Place the pans in oven and bake until a toothpick inserted into the center of the cakes comes out clean, about 35 minutes. Allow the cakes to cool for at least 10 minutes before removing them from the pans. Let the layers cool completely before assembling the cake.
Place one of the layers on an 8-inch cardboard cake round or a freezerproof cake plate. Using a rubber spatula, evenly spread 1 quart of the softened ice cream over the top of the plated layer. Place the remaining cake layer on top of the ice cream. Put the cake in the freezer to harden, about 1 hour.
Transfer the remaining softened ice cream to a mixing bowl. Cream by hand with a rubber spatula until it reaches the consistency of frosting. Frost the top and the sides of the cake with the ice cream.
Note: If the ice cream becomes too soft to work with, place both the bowl and the cake in the freezer until the ice cream is firm, about 30 minutes. When the cake is frosted, transfer it to the freezer to harden, about 1 hour. Serve immediately upon removal, using a warm knife to slice.