Take some examples of varying severity. In front of the class say,

I have identified many examples of plagairism on the last assignment. Let me show you some...

Then say, "I have identified more than X cases. If you think you may have been one of them, let me know by email, and I'll let you redo the paper and ignore your first submission.

Anyone that I have identified as a plaigairist who does not offer to redo your paper will be failed. If you think you may have quoted too liberally from an unacknowledged source, I would strongly recommend playing it safe and redoing your paper.

Then on the next class come in, pick a random name out of the list of people who have not reached you yet, call that person by name, and ask him or her if he or she wishes to fail the course. After the kid says no, say, "Then are you going to be redoing your paper?" Then to the rest of the class say, "If you should redo the paper and haven't talked with me yet, I suggest doing so soon."

I believe that once expectations are clearly enough set, most of the problem will disappear fairly easily. As for the rest? I have absolutely no problem with failing cheaters as fast as possible.