Preheat oven to 350°F (177°C).
Lightly butter or oil a 13-1/2" x 8-1/2" baking dish.
Cook contents of rice and seasoning packets with water and 1 tablespoon butter or margarine according to package directions.
In a large saucepan, bring 6 cups water to boiling.
Add cabbage.
Simmer until leaves are pliable, about 3 minutes.
Drain.
Remove outer cabbage leaves (about 24 leaves).
Cut out hard center rib at bottom of each leaf.
Melt 2 tablespoons butter or margarine in a medium skillet.
Add celery, onion, carrot, corn and mushroom pieces.
Cook, stirring constantly, until celery and onion are tender, not brown.
Stir vegetable mixture and cheese into cooked rice.
Place 2 to 4 tablespoons of rice mixture on each cabbage leaf, depending on size of leaf.
Roll up, tucking sides in during rolling.
Arrange rolls in prepared baking dish, folded side down.
Cover with aluminum foil, crimping it tightly to edges of dish.
Bake 25 minutes.
Combine soup and milk in a medium saucepan.
Heat and serve with cabbage rolls.