A slightly different answer. Though of course, I agree with the others that say you should check your institution's policies before deciding whether / how to handle this on your own.
On the syllabus...I...stated that a student who plagiarizes once will receive a zero for the assignment. A student who plagiarizes twice...may withdraw or take an F for the course.
This is a very reasonable policy. You should have stuck with it.
a student...asked for the chance to submit another essay in its place. With some misgivings, I agreed to this.
Mistake! What's the point of having a policy if you're not going to follow it? I hope you at least (1) did not agree to give her full credit, and (2) will file a report so that she cannot have a new "first offense" in a different class.
As I was grading the journals due that day, I found that her new work was also created by AI...she apologized via email, "explaining" that she had submitted the journal [before being caught on] her essay.
Your policy probably didn't envision this scenario -- you may feel that a student who submits two plagiarized assignments before being caught deserves less of a penalty than a student who submitted one, got caught, and then submitted another anyway. Or you may feel that plagiarism is so manifestly unacceptable that the timing doesn't matter. Either way, make sure your policy accurately reflects what you want to happen.
At this point, I can either still let her rewrite the essay and warn her yet again, or tell her that under the circumstances, I cannot accept a rewrite from her
The essay case is already adjudicated, you already agreed to let her rewrite that one. But this journal case is new and separate. As I see it, there are two options:
- You could treat this journal case as a first offense (since you've already agreed to overlook the essay case) and assign a zero on the journal. Or,
- You could treat this journal case as a second offense (since she did in fact plagiarize twice) and allow the student to withdraw or take an F for the course.
Personally, I would lean toward the former if she had voluntarily confessed about the journal. Since she didn't, I would lean toward the latter. But either is defensible at this point.