What to do about a student whom I suspect of having plagiarized but who claims not to have?
Inform the student of the school's official procedure for appealing an accusation of plagiarism, and suggest he follow that procedure. (Also see: How can I prove that I didn't plagiarize.)

Policies and procedures are there exactly for situations like this one.

Generally if the student chooses to appeal, the procedures involve other parties (a dean, or a committee dedicated to such matters, for example) evaluating the student's side of things and your side of things (your reasons for believing that the student plagiarized) and then deciding on next steps according to school policies. (Here's an example.) So if you think the student is likely to appeal, you can start to prepare your side of things - the "evidence" - in the meantime.  (Personally, when in this situation I also like to let the department head know that this appeal might be coming.)

Note that you don't need to have caught the student on videotape, or anything like that. You'll tell the committee what you know, they'll make the decision as to whether it's enough evidence, and they'll make the decision as to whether the consequences you have imposed should stand. Deciding whether your evidence meets "standards of proof" for an appeal according to school policy is their job, not yours (they have much more experience than you in this matter.) Assuming you made your accusation in good faith, you don't need more evidence for an appeal than you did for the original accusation.