In a large, heavy saucepan, bring the red wine, Port, sugar, honey and cinnamon stick to a simmer and cook until the sugar is dissolved. Cut a small "x" into one end of each apricot with a paring knife. Bring a pot of water to a boil. Poach the apricots for 30 seconds. Remove with a slotted spoon to a colander and slip the skins from the fruit.
Add the apricots, adding more wine as necessary to cover the apricots, and simmer covered, turning occasionally, until tender, about 15 minutes. Remove from the heat and divide the apricots among 4 bowls. (If desired, cut in half first and discard seed.) Bring the poaching liquid to boil and reduce by half. Discard the cinnamon stick. Let the syrup cool slightly.
Place 1 scoop of ice cream over each apricot serving, and top with poaching liquid. Sprinkle toasted almond slivers over each serving, and serve immediately.
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