Preheat the oven to 350°F. Spray a 12 cup Bundt pan with Baker's Joy (after making a ton of Bundt cakes, I've had 100% success with Baker's Joy, so I'm mentioning the brand).
Place the cake mix, melted ice cream, and eggs in a large mixing bowl. Mix well, then beat until smooth. Pour into the Bundt pan.
Bake the cake until it springs back when lightly pressed with your finger and just starts to pull away from the sides of the pan, 38 to 42 minutes. Remove the pan from the oven and place it on a wire rack to cool for 20 minutes. Run a long, sharp knife around the edge of the cake and invert it onto a small rack, then invert it again onto a second rack so that the cake is right side up to complete cooling, 30 minutes more.
To make the frosting, beat together the cream cheese and butter. Gradually add the sugar and beat until smooth. Beat in the vanilla. Put it all in a heavy zipper bag (or an icing bag fitted with a round tip if you have one), snip off the bottom corner of the bag, and pipe it up and down the sides of the cake.