I'd suggest tip rooting them. You take the end of one of the vines (one without berries on it) and put it in the dirt by an inch or so. Then put a rock on top of the vine to keep it there. Wait a month or two, then cut the vine off. You can make the process much easier by putting the tip directly into a pot of soil, so you just pick up the pot after cutting the vine. You could also mark them and dig them up in the fall/spring when they are dormant. The above aside, you may want to consider buying a named variety. My understanding is that the fruit size is much larger, and there may be some disease resistance bred into them. For a comparison picture (from BackyardBerryPlants), see below. The ones on top are Jewel and the ones on the bottom are wild black raspberries.

I got my teeny CJ from Henry Fields, as a matter of fact! It was really clearance priced. It was incredibly healthy, just quite small. I do recommend them for the little starts of these as far as health of the little plant goes. They might have different sizes. I'm sure mine said it was in a 3 inch pot, so I knew what I was in for that way. If you do go with the little babies, just make sure to pay more attention to it than you might for a bigger plant. The roots aren't as forgiving since there aren't too many of them comparatively ;) I potted mine up in good soil and checked frequently for water needs. It was still really easy to grow the whole time. I've had zero problems.

Sorry I have to update to the 2004 AEN, had ordered 2 over a month and still waiting to receive them.

Please use the new listing, Thanks