Adjust rack to center of the oven and preheat oven to 350 degrees. Butter two 9-inch round layer-cake pans, line the bottoms with baking-pan liner paper or wax paper cut to fit, butter the paper, dust with flour, then invert over a piece of paper and tap lightly to shake out excess. Set aside.

Place the chocolate in the top of a small double boiler over hot water on moderate heat. Cover until partially melted, then uncover and stir until completely melted. Remove from the hot water and set aside, uncovered, to cool slightly.

Sift together the cake flour, baking powder, baking soda, and salt and set aside.

In the large bowl of an electric mixer cream the butter. Add the sugar and beat to mix well. Add the eggs one at a time, beating until the egg is thoroughly incorporated after each addition. Mix in the vinegar. The mixture will look curdled-it is O.K. Add the melted chocolate and beat only until smooth.

On low speed add the sifted dry ingredients in three additions, alternating with the milk in two additions. Scrape the bowl with a rubber spatula and beat only until smooth after each addition.

Place half of the mixture in each prepared pan and smooth the tops.

Bake for 35 to 40 minutes until the layers begin to come away from the sides of the pans and the tops spring back when lightly pressed with a fingertip.

Remove from the oven and, with a small, sharp knife, cut around the insides of the pans to release. Then let the layers stand in the pans for 5 minutes.

Cover each layer with a rack, invert, remove the pan, peel off the paper lining, cover with another rack and invert again, leaving the layer right side up to cool.

Prepare a large, flat cake plate by lining the sides with four strips of wax paper. Place one layer upside down on the plate, checking to be sure that the papers touch the layer all around. If you have a cake-decorating turntable or a lazy Susan, place the plate on it. Prepare the icing.

Place the chocolate and the butter in the top of a small double boiler over hot water on moderate heat. Cover until partially melted, then uncover and stir until completely melted.

Meanwhile, place all the remaining ingredients in the small bowl of an electric mixer. Beat briefly only to mix. Set the small bowl in a large bowl and fill the empty space left in the large bowl with ice and water, filling to about three-quarters the depth of the large bowl. (If you are using an electric mixer on a stand, use the large mixer bowl for the ice and water but adjust the stand for "small bowl.")

Add the melted chocolate and butter, and beat until the mixture thickens slightly. Remove both bowls (together) from the mixer. With a rubber spatula stir the icing over the ice and water until it thickens to the consistency of thick mayonnaise.

Spread a scant third of it about 1/4 inch thick over the bottom layer of cake. Cover with the other layer, placing it right side up (both bottoms meet in the middle). Spread the sides and the top with the remaining icing.

It may either be spread smoothly with a long, narrow metal spatula, or it may be formed into swirls.

Remove the strips of wax paper by gently pulling each one out toward a narrow end.