Originally posted by sweetpea:
This sounds like it might be anthracnose:
11e. Small, circular, tan to brown spots on mature or nearly mature fruit. Lesions expand rapidly, with a tendency to form concentric rings, and may or may not be sunken. Individual lesions may reach a diameter of 4-5 cm, but infection can be more extensive when lesions coalesce. Lesions are firm to the touch but typically develop orange to pink, slimy spore masses in their centers. ANTHRACNOSE (Colletotrichum acutatum; Colletotrichum gloeosporioides)
Do NOT use lime sulfur spray on apricots, although it is used on other stone fruits for this problem as a dormant spray.
There is a toxic spray that will kill this, which I won't list here. And I'm not finding an organic remedy for this, but I haven't had time to really look. Maybe you can find one.