In a large saucepan cook potatoes, carrots, celery, and onion in 1 cup boiling water, covered, for 12 to 15 minutes or until tender. Do not drain.
Meanwhile, in a small saucepan melt butter; stir in flour until smooth. Add milk all at once. Cook and stir over medium heat until mixture is thickened and bubbly. Stir into vegetable mixture; stir in ham and cheese. Stir until cheese melts and mixture is heated through.