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Crack 10 of the apricot pits to reveal an almond-like nut within. Reserve these nuts, discarding the remaining pits. In a large unlined copper jam pot or a large, heavy stock pot, combine the apricots, reserved nuts, and sugar. Stir to dissolve the sugar. Cook over moderate heat, stirring regularly, for 1 hour. Do not allow the mixture to burn or to stick to the bottom of the pan. The mixture will turn very thick and bright orange and most of the apricots will melt into a purée. Transfer to a bowl and set aside at room temperature for 24 hours.
When ready to complete the jam, prepare four 8-ounce canning jars with lids by sterilizing them in boiling water according to the jar manufacturer's instructions.
Reheat the mixture over moderate heat, until very thick. Transfer to the hot, sterilized jelly jars, leaving room at the top. Seal according to the jar manufacturer's instructions. Store in a cool, dry, place for up to 1 year.
This will make a rather tart jam. For a sweeter spread, add an additional 1/2 cup of sugar.