Preheat oven to 350F. Line the bottom of a 9-in cake pan with a parchment paper round and lightly grease bottom and sides. Set aside.
In a large bowl, whisk sugar, oil, egg, milk, coffee, and vanilla until smooth. Add flour, cocoa, baking soda, baking powder, and salt to liquid and gently fold until just combined (batter will be slightly thin)
Scrape batter into prepared pan. Bake cake at 350F 28-30 minute or until cake comes out with moist crumbs sticking to it. Cool cake completely on a wire cooling rack then, invert onto a serving plate and cool completely.
Line a 9-in cake pan with plastic wrap and carefully spoon softened ice cream into bottom of it, flattening into cake pan. Cover top with plastic wrap and freeze ice cream until completely solid, about 2 hours.
Remove plastic wrap from ice cream and invert cake onto ice cream. Cover with plastic wrap and freeze cake 2 hours or until frozen solid. Store cake in freezer.