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1
In a small saucepan over medium heat, simmer water, sugar, and basil leaves until sugar melts and basil goes limp, about 5 minutes. Turn off heat and let sugar-water mixture steep 15 minutes.
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2
Meanwhile, halve the apricots by slicing along their dimple all the way around. Remove the pits, which should slip out easily. Slice each half into 2 or 3 pieces. By now the basil mixture should be finished; remove basil leaves, pressing against the side of the pan with the back of the spoon to coax all the flavor out of them. Discard basil.
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3
Transfer apricot pieces into saucepan containing basil-infused syrup. Return heat to medium, and cook until apricot pieces are soft and tender, about 8 minutes. Set pan aside and bring to room temperature.
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4
If you have a hand blender, use it to puree the apricot mixture right in the pan until smooth. Alternatively, puree apricots in blender or food processor. Transfer apricot mixture to bowl of double boiler or, if you don't have one, transfer apricot mixture to a metal bowl, and add egg yolks one at a time, mixing well after each addition. Set bowl over a small saucepan with a couple inches of water in it, turn heat to medium, and begin heating apricot mixture over water, stirring thoroughly and often. After about 7 minutes over the simmering water, apricot mixture should begin to thicken. It will only thicken slightly, never getting as viscous as creme anglaise or pudding. The whole thing should be done after 10 minutes. Remove from the heat and stir in lemon juice.
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5
Pour cream into large bowl and set strainer over the bowl. Pour apricot custard through strainer into cream. Press on any solids left behind, then discard. Stir custard and cream together until they turn a uniform shade of light orange. Chill thoroughly.
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6
Process ice cream in ice cream maker according to manufacturer's instructions.
