Preheat oven to 350°. Grease and flour a 10-inch Bundt pan. Set aside.
Combine cake mix, pudding mix, 1 cup water, eggs, and oil. Using an electric mixer on low speed, mix until combined. Increase speed to medium and beat for 2 minutes, scraping down sides as needed.
Transfer 1 & 1/2 cups of cake batter to another bowl. Stir in chocolate syrup. Pour light batter into pan. Top with chocolate batter. Using a knife or metal spatula, gently swirl through the batters to marble.
Bake for 55-60 minutes, until a toothpick inserted near the center comes out clean. Cool in pan on wire rack for 15 minutes. Remove cake from pan and cool completely on wire rack.
Combine chocolate and butter in a small saucepan. Cook over low heat, stirring constantly, until melted. Remove from heat and stir in powdered sugar and hot water. Add additional hot water, 1 teaspoon at a time, if necessary to attain drizzling consistency. Drizzle icing over cake.