Warm the milk, melt the chocolate in it and dissolve the sugar with a pinch of salt. Pour the mixture into a chilled, tall container, add the chilled mascarpone and chilled coconut milk and mix vigorously once. Pour the mixture into the ice cream machine and let it run for 60 minutes.
If you like, you can let the chocolate chips trickle in slowly after 30 minutes in the ice cream machine.