Preheat oven to 350°F. Grease a bundt cake pan generously with nonstick baking spray.
Melt the chocolate in a double boiler over simmering water, stirring until smooth. Set aside to cool slightly.
Pour the beer into a liquid measuring cup and set aside. (This will release some of the carbonation).
Sift together the flour, baking powder, baking soda, and salt together in a medium bowl.
In the bowl of an electric mixer, beat together the butter and sugar until light and fluffy. Add the eggs one at a time, beating after each. Mix in the vanilla and melted chocolate. Scrape down the bowl often to make sure batter is mixing evenly.
Add the flour mixture in 3 additions, alternating with the stout beer, and ending with the flour mixture. Scrape down bowl as needed. Mix on medium speed for a few seconds until smooth and creamy.
Pour batter into prepared bundt pan and bake until a toothpick comes out clean, about 50-60 minutes. Leave the cake in the pan for 15 minutes and then turn out onto a wire rack and cool completely.
To make the icing, heat the cream over medium high heat or in microwave until it starts steaming. Pour the cream over the chocolate in a heat proof bowl. Let stand for 5 minutes and then stir until the mixture becomes silky smooth. Add stout beer.
Let icing cool until it starts to thicken, then pour over cooled bundt cake. Let icing set up for 20 minutes before serving.