Preheat the oven to 350 degrees F (175 degrees C). Grease and flour a 9 or 10 inch Bundt pan.
In a large bowl, mix together the cake mix, eggs, 1/2 cup of sugar, oil, 3/4 cup of apricot nectar and brandy using an electric mixer on low speed. Mix for 3 minutes on high speed. Pour into the prepared pan.
Bake for 45 minutes in the preheated oven, or until a toothpick inserted into the crown of the cake comes out clean. Cool in the pan, then invert onto a serving plate.
To make the icing, combine 3 tablespoons of apricot nectar, 3 tablespoons of sugar and butter in a saucepan. Cook over medium heat, stirring, until butter is melted and sugar has dissolved. Let cool one minute, then pour over the cake.