In a large bowl, combine apricots, rosemary and sugar; mix well and let sit about 30 minutes until juices form at the bottom of the bowl.
Transfer the mixture, with juices, to a large pot and heat over medium high.
Once mixture begins to boil, reduce heat to medium and cook, while stirring often, until apricots are soft and begin to lose their shape, about 30 minutes, or more (they should not have darkened significantly).
Add a little water (up to a cup) if mixture starts sticking.
Press apricots through a ricer or fine mesh strainer and return to pot; discard solids.
Taste for sweetness and add sugar if needed; cook jam a few minutes more to dissolve any added sugar (flavor mellows as it cools).
While still hot, ladle jam into hot, sterilized canning jars, leaving about 1/4 inch head space at the top; remove air bubbles by running a nonmetallic utensil or skewer between the jar and the jam.
Top with new, clean lids, close tightly, and let come to room temperature.
Check seal by pressing down in center of lid, it should not spring up.
Process any jars that have not sealed in a hot water bath for 15 minutes; retest.
NOTE: To sterilize the empty jars, fill a large pot with water and bring to a boil over high heat, then reduce the heat so that the water is barely bubbling. Have ready seven 8-ounce canning jars with 2-piece lids. Immerse the pint jars in the canning kettle. Place the rings and lids in a separate small saucepan and cover them with hot water. Leave the jars and lids immersed while you cook the jam.