Wash, trim, and scrape the beets. Dice them into a saucepan and add the stock. Bring to a boil and simmer, covered, for about forty-five minutes.
Strain the liquid and discard the beets. Stir in the sugar and lemon juice. Chill.
Peel the cucumber, halve it lengthwise, and remove the seeds. Cut the cucumber into fine dice or into very thin strips. Add to the soup. Add salt to taste. Ladle the soup into cups, top with sour cream, and sprinkle with dill.