Bake the cake in 2 layers as directed on package. When cool, slice each layer in half horizontally, making 4 layers. In a medium bowl, combine the 2 cans of sweetened condensed milk with the fresh lemon juice. Reserve half of the mixture and set aside.
Use half of lemon milk mixture to put between layers. With the reserved mixture, gently fold in the whipped topping, and use to frost the entire cake. Chill overnight before serving.