Slice the cake in half; split one half into small pieces and set aside. Slice the other half into 12-14 thin slices; place it 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 soft. Softly press the small cake pieces into ice cream. Wrap and freeze. Just before serving, cut strawberries and sweeten to taste. Slice dessert into squares and top with strawberries.