Preheat the oven to 375℉ and place a baking sheet on the lowest rack to catch any drips as the cake bakes on the middle rack. Grease a tube pan. You can line the bottom of the pan with parchment paper to help remove the cake later.
In a 2- to 3-quart pot, mix together the sugar, unsweetened chocolate, butter, and 1 cup of water. Place over medium heat and stir occasionally until all of the ingredients are melted and blended. Remove from the heat and let cool slightly.
Meanwhile, sift together the flours, baking soda, baking powder, and salt. In a small bowl, stir together the milk and vinegar. (Alternatively, you can use buttermilk.)
When the chocolate in the pot has cooled a bit, whisk in the milk mixture and eggs. In several additions, whisk in the dry ingredients. When the mixture is smooth, add the vanilla and whisk once or twice to blend.
Pour the batter into the tube pan and bake on the middle rack until a skewer inserted in the center comes out clean, about 30 to 35 minutes. Let the cake cool for 10 minutes, then remove from the pan and cool on a rack, then let cool completely.
Meanwhile, melt the chocolate chips in a double boiler, then let cool to room temperature.
In a large bowl, using an electric mixer, whip the heavy cream with the chocolate to make a frosting.
When the cake is cooled, frost the top and the sides with the frosting.