Preheat the oven to 350 degrees F (or what the box of cake mix indicates) and grease 3 8-inch cake pans well.
Make the cake mix according to the package directions.
Divide the prepared batter evenly between the 3 prepared pans and bake according to package directions, it should be about 20-25 minutes.
While the cake is baking line 2 more 8-inch cake pans with plastic wrap, pushing it down into the bottom of the pan.
Divide the softened ice cream evenly between the two pans, using a spatula to press it into the bottom and corners well (this helps it look nice), smooth out the top and place it in the freezer to harden.
When the cakes come out of the oven allow them to cool for 10 minutes in the pans and then remove them carefully and allow them to fully cool.
When you are ready to assemble your cake, add your heavy whipping cream, sugar, and vanilla to the bowl of your stand mixer and beat on high speed until light and fluffy and soft peaks form, about 3 minutes.
Place a layer of cooled cake on a serving plate or cake stand.
Remove one layer of ice cream from the freezer and use the edges of the plastic wrap to pull it from the pan. Flip the ice cream over and place it on top of the cake. Peel off the plastic wrap. Repeat with the next cake layer, the last ice cream layer, and the final cake layer.
Spread the whipped cream evenly over the cake and add sprinkles if you'd like.
Place the cake in the freezer for at least 4 hours or overnight before serving.
To serve allow the cake to rest at room temperature for 30 minutes before slicing and serving.
Notes
If you don't have more than 3 cake pans you can just use the pans that you baked the cake in, once they are cooled, for the ice cream part. Just plan on more time to allow the pans to cool etc.
You don't have to add the whipped cream icing but it looks pretty. Sometimes I don't add it and I'll just drizzle the top with chocolate sauce before serving.
If you want to make this cake ahead of time, do the whole thing and get it into the freezer. After a few hours the outer whipped cream later will have frozen and gotten hard. Remove the cake from the freezer, wrap in 2 layers of plastic wrap so it's well covered, and keep in the freezer until you are ready to serve, up to 2 weeks.