Cut the cake in half; tear one half into small pieces and set aside. Cut the other half into 12-14 thin slices; arrange in the bottom of a waxed paper-lined 13-in. x 9-in. dish, overlapping as needed.
Spread softened ice cream over cake, pressing down to smooth. Gently press the small cake pieces into ice cream. Cover and freeze. Just before serving, slice strawberries and sweeten to taste. Cut dessert into squares and top with strawberries.