Writing a recommendation letter for a student I reported for academic dishonesty
The best option is: Don't write the letter. In a case like this, I would outright refuse to provide a recommendation.
However, if you are not comfortable with a non-negotiable refusal, you might want to simply tell the student that you cannot provide a positive letter for them. If the student insists, then you should write an honest letter that gives a balanced evaluation of their ability and personal traits. The fact that you caught the student cheating is a critically important piece of information in this regard, and it needs to be included. The cheating will almost certainly be the most salient feature to anyone who reads the letter, which will make the net effect of the letter, regardless of whatever else you say in it, strongly negative. And if the student insists on you writing the letter in this situation, that is basically what they deserve.
TL; DR: The student probably wants you to write a letter to mitigate the harm of the incident because they already expect admissions committees to know about it. In that case, mention it briefly and move on to your standard letter.
There are only three reasons I can think of why a student would ask you to write a letter in this situation. How you respond depends on the reason.
Despite, or perhaps because, of the incident you have become a mentor to the student and genuinely know the student very well. The student knows that they have regained your trust and confidence, and that you're in a good position to give them a solid recommendation.
A. If this is the case, and you truly do now trust the student and want to offer a strong recommendation, you can mention that the two of you got off to a rocky start, and allude to the issue without going into detail (you don't need to use the words "misconduct" or "cheating"). You could say that the student worked hard to regain your confidence, and that you are now happy to strongly recommend etc.
B. If this is only sort of true, and you don't think very highly of this student and/or have reason to think they might cheat again given the right circumstances, you should decline to write the letter.
It doesn't sound like your situation falls into the first category, however, which brings us to the second possibility.
The student knows that this incident will appear on their record, and that their prospective programs will see it. They believe they have at least somewhat redeemed themselves in your eyes, and are hoping that a generally positive letter from you will mitigate the harm of the incident. I suspect that this is what's actually going on. If so, you should confirm this. Ask the student specifically whether the incident will already be known to admissions committees.
A. If the answer is yes, write whatever letter you would have written absent the misconduct but with an additional paragraph noting that you were involved in the incident and believe that the student has faced sufficient consequences, and that your letter focuses on their performance in the subsequent class. If you have reason to think the student has learned from the incident and it's not likely to be repeated, include that in the paragraph acknowledging the misconduct, but don't dwell on it.
B. If the answer is no, the misconduct is sealed or otherwise unlikely to become know to prospective programs, we come to the third possibility.The student is extremely naive, and doesn't realize the potential harm you could do to their admissions chances by writing about the misconduct.
A. In this case, absent any particularly strong admiration for the student, I would decline to write the letter. At the very least, you need to strongly caution the student about how bad a letter from you could be for them.
B. If you really still want to write the letter and your student still wants you to write it after being warned: You could take either approach above, depending on what feels more comfortable to you—either allude to the issue in the most general terms, or mention it but state specifically that it is not the focus of your letter. You should definitely warn your student about this beforehand, though, and perhaps allow them to see the letter before you send it.
To what extent has the student regained your trust since the incident?
As other answers say, the incident is too significant to ignore. However, what you write in the letter should be not just “X happened”, but “X happened, and here’s how it affects my judgement of the student now.”
Most probably, your trust in their future honesty is still shaken, in which case (if you’re honest about this) your letter will carry a significant negative impact, to the extent that you should probably decline the request.
However, there could be mitigating circumstances that make you genuinely confident that the student will not re-offend. For instance — since you mention they were a newly-arrived international student — perhaps their home country had a different culture around expectations for homework, but you feel confident that they have now absorbed US academic norms. In that case you can say so in the letter:
I would be remiss not to mention that [incident occurred]. However, I feel this should not be held against X’s future prospects, since [mitigating circumstances, and why you feel they’ve regained your trust].
I would expect, for most readers, this would still have some negative effect on their assessment of the student — but much less than if they learned of the incident some other way (e.g. mentioned on the student’s academic transcript) without an explanation of the mitigating factors. So overall, submitting this letter would be a positive thing (provided that the rest of it is enthusiastic).
Disclaimer: I have only a little experience of admissions/hiring; I hope more experienced users can confirm (or differ with) my expectation of how such a letter will be received.