Make the cake: Prepare the cake mix as the label directs for a 9-by-13-inch cake. Bake; cool slightly in the pan, then invert onto a rack to cool completely.

Clean out the cake pan and line it with plastic wrap. Cut the cake in half lengthwise, then in thirds crosswise to make 6 rectangles. Using a serrated knife, slice each rectangle in half to make two layers. Arrange half of the pieces snugly in the cake pan. Crumble the remaining pieces and press tightly into 6 small balls; arrange 1 cake ball in the center of each cake rectangle in the pan.

Make the mousse: Heat the chocolate, butter, coffee and 1/4 cup water in a heatproof bowl set over a saucepan of simmering water (do not let the bowl touch the water), stirring, until melted. Remove the bowl from the pan; stir until cool. Reserve the simmering water. Fill a large bowl with ice water.

Whisk the egg yolks, 1 1/2 cups sugar and 2 tablespoons water in a separate large heatproof bowl. Set the bowl over the saucepan of simmering water and whisk until pale yellow, 6 to 8 minutes. Add the chocolate-coffee mixture and the vanilla and whisk until combined, about 2 minutes. Remove the bowl from the pan and set in the bowl of ice water; whisk until slightly cool but not thick, about 4 minutes.

Beat the egg whites and salt in a bowl with a mixer until foamy. Add 2 tablespoons sugar and beat until almost stiff; gently fold into the chocolate-yolk mixture to make a dark chocolate mousse. Spread 5 cups over the cake and cake balls. Freeze until firm on top, about 30 minutes.

Beat the heavy cream and the remaining 4 tablespoons sugar until soft peaks form; fold into the remaining mousse. Remove the cake from the freezer; spread with the light chocolate mousse, cover with plastic wrap and freeze until firm, at least 6 hours or overnight.

Make the chocolate shell: Stir the chocolate, heavy cream and 4 tablespoons corn syrup in a heatproof bowl set over a saucepan of simmering water (do not let the bowl touch the water) until melted. A spoonful at a time, spread half of the shell mixture over the frozen mousse. Return to the freezer.

Make the chocolate sauce: Add the remaining 1 tablespoon corn syrup and the butter to the remaining chocolate shell mixture. Microwave 30 seconds, then stir until glossy.

To serve the cake, beat the heavy cream with a mixer until foamy. Add the sugar and beat until soft peaks form. Remove the cake from the freezer; invert onto a baking sheet and unmold, then invert again onto a platter, chocolate shell-side up.

Cut the cake in half lengthwise using a warm knife, then cut into thirds crosswise to make 6 rectangles; cut each rectangle in half diagonally to make 2 triangles. Top each triangle with whipped cream, ice cream and the prepared chocolate sauce.

Photograph by Kang Kim