I speak as an undergraduate here, so I realise that there may be some things about your position that I can't really appreciate properly.
I also think that if you made it very clear that NO collaboration was allowed then it is true, this student cheated.
However, I would say that a no collaboration rule is something you may want to be wary of if you're keen to have a good class ethos. I have always found that classes where some level of conferring between students has seemed to bring the best out of the assigned task. Direct copying and working together is obviously outside this, here I am for instance talking about doing your work at a computer next to someone doing the same work - and discussing the task with them as you go through. The very best lecturers I've had, whose coursework I have learned the most from, have been the ones who allowed and encouraged this practice.
The end result is a class ethos and atmosphere where people learn from one another and take an active interest in the subject matter. Another result is that people can become more competitive among themselves, especially if the coursework is creative enough to allow some 'flair' to take place. (For instance in a programming task 20% of the marks are often given for extensions). A side effect of this competition is that students will not tolerate other students stealing from them. They certainly won't tolerate copying from previous years solutions. Quite apart from this intolerance of cheaters the sense of competition allows a sense of personal pride, whereby students want to hand in a good piece of original work.