Combine cucumber, onion, celery, green pepper, and parsley in container of an electric blender; process until smooth. Set mixture aside.
Melt butter in a large saucepan over medium heat; add flour, and cook 1 minute, stirring constantly. Add chicken broth and half-and-half; cook, stirring constantly, until mixture is thickened and bubbly. Stir in salt and pepper.
Add reserved cucumber mixture to sauce; cook over medium heat, stirring frequently, until thoroughly heated.
Ladle soup into individual serving bowls; garnish each serving with dill weed, and serve warm. Soup may also be served chilled.