How to handle a student's public speaking anxiety when teaching a class that requires presentations?

In one of the courses I'm teaching this semester, each student has to give a 20-minute in-class presentation of a paper relevant to the topic of each class. One of the students has approached me asking to be excused for this requirement, on the grounds that she is terrified of having to speak in front of others. She is willing to do an alternative assignment that doesn't involve a presentation; more significantly, I think, is the fact when I pressed her a bit about this, she seemed willing to straight up forgo this assignment and get a zero grade for this particular part of the course. Conversations I've had with other faculty who have had her in previous courses confirm that she is, in fact, a good enough student that she should be able to properly understand the kinds of papers we are reading in this course. So, this seems to be a genuine anxiety problem.

Given that I had my own mental issues way back when I was an undergrad, I know that a just-suck-it-up type of attitude on my part is likely to cause more problems than it solves. I'm willing to accommodate her request for an alternative assignment. On the other hand, there are two reasons why I'm in principle against this decision. First, it is unfair to the rest of the students (and also, it might get me a reputation as a lecturer that students can sway). Second, giving her an extra assignment effectively amounts to kicking the can down the road: the more she delays dealing with her problem, the harder it will become to solve, and chances are that the kind of environments she will encounter after graduation will not be as speaker-friendly as my class is.

How would you proceed here?