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Melt 3 TBS butter in an 8" X 8" pan.
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With a small piece of wax paper, butter the sides as well.
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Add the syrup.
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Remove from heat and sprinkle in the rum.
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Divide apple into eighths, core and arrange in neat rows in syrup mixture.
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Cream 1/3 cup butter, add 1/2 cup sugar and beat until light.
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Beat in egg yolks, one at a time.
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Sift together dry ingredients and add alternately to creamed mixture with milk.
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Stir in grated apple.
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Beat egg whites stiff.
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Beat in remaining 1/4 cup sugar, and fold gently into batter.
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Spread batter evenly over apples in pan.
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Bake 350*F oven for 40 to 45 minutes.
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Let stand 5 minutes, then invert onto a flat cake plate.
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Serve hot or warm with whipped cream.