Carefully cut 1 in. (2.5 cm) horizontally off top of cake with serrated bread knife, keeping the round in one piece. Set aside. Carefully hollow out bottom portion of cake, leaving 1/2-in. (1-cm) thick shell (removed cake pieces can be frozen and used for recipes such as trifle or rum balls).
Fill cake hollow with ice cream. Put top of cake back in place. Wrap and freeze until ice cream is firm, about 1 hour.
When ready to serve, mix strawberry slices with sugar. Slice cake and serve with strawberries on side and top with whipped cream, if using.