A picture might be good if you're still having the issue. Have you tried setting the melons on something (besides the ground) as soon as they start forming?

Black rot?

http://www.hort.uconn.edu/ipm/veg/htms/blkrtcuc.htm

Watermelon.
Spots start as small, round, dark greenish-tan to black water-soaked areas.
They enlarge slowly, becoming brown to black in the center.
They remain smooth for a long time, but become depressed as they enlarge.
The spots are usually surrounded by a narrow water-soaked area with an irregular border.
Under moist conditions, tiny pale and dark dots develop in the center of the spots, although these can be difficult to see.
The spots may become leathery, the centers become darker, and cracks may radiate from the center of the spots.
When humidity is high, a white cottony fungal growth may grow over the spots. Gummy ooze is usually absent.
If the fruit is sliced open, cutting the spot in half, the rind is dark brown to black directly below the spot, and progressively lighter toward the edges.

Here's a pic I found on a pumpkin:
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