Cut potatoes into small pieces. In a 3 quart saucepan, boil potatoes until extremely tender. While this is happening, wash asparagus. Remove woody ends. Cut into 1 inch sections, setting aside tips. Add non tip sections to potato pot and cook until asparagus has gone beyond bright green to olive green, but not brown. Add dill.
Asparagus should be soft enough to mash along with potatoes, in the cooking water. Don't throw out any of the cooking water. Lightly brown flour in a small saucepan and allow to cool somewhat before adding milk. Add liquid gradually to make a smooth roux and add the roux to the soup. Add the asparagus tips and continue cooking the soup just until they are tender and bright green. Salt to taste.
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