Preheat the oven to 350 degrees F.
Grease and flour the sides of an 8-inch springform pan, and line the base with greaseproof or parchment paper.
In a large bowl or in an electric food mixer, cream the butter until soft. Add the sugar and beat until the mixture is light and fluffy.
Whisk the eggs and vanilla extract together in a small bowl. Gradually add the eggs to the butter mixture, beating constantly. Sift in the flour and baking powder, and gently fold into the butter mixture. Add the milk to the mixture, and mix gently to combine.
Tip half of the cake mixture into another large bowl, and into this bowl, fold in the sifted cocoa powder.
Place the cake mixtures into the prepared pan by alternating spoonfuls of the vanilla batter with the chocolate batter, and then with a skewer or knife, gently draw swirls through the cake mixture to 'marbleize' it. Don't overmix or you won't have that wonderful marble effect. Bake in the oven until a skewer inserted into the center of the cake comes out clean, about 45 minutes. Turn the cake out onto a wire rack and cool completely.