Three years ago I planted three Prime-Jim blackberry bushes in my garden, and my reason for selecting this variety is that it produces two crops every year, the first in June and the second in late September and into October. The picture below was taken on October 10, 2013.
At the time, the vendor I choose did not carry the Prime-Ark 45 which is much sweeter berry, but I hesitate to recommend it because the University of Arkansas has now developed a new variety called Prime-Ark Freedom.
While plants are currently unavailable, seven propagators are licensed to grow Prime-Ark Freedom so look for them in 2014. Here you will find a complete list of all propagators of Arkansas varieties including Prime-Ark 45, Prime-Jim, and Prime-Jan.
Information on growing blackberries is availble as a PDF file from Missouri State nursery in Mountain Grove. Good Luck and Good Gardening!