Wash the asparagus. Peel the white asparagus completely, the green only in the lower third and cut off the woody ends in each case.
Bring the stock to the boil in a saucepan and simmer the asparagus peelings in it for 20 minutes, just below boiling point. Pour the broth through a sieve into a pot, squeezing the asparagus peels well with a ladle and removing them.
Cook the chopped potatoes in the soup until soft. Cut the asparagus spears diagonally into slices about 1/2 cm thick, add to the asparagus stock and cook for 5 to 10 minutes just below boiling point until al dente. Remove the asparagus tips from the stock and set aside for the garnish. Add the cream to the asparagus stock in the pot and heat to just below boiling point. Remove the asparagus soup from the heat - season with salt, pepper and lemon juice. Add the butter and stir in with a hand blender.