Do you have a car? I think that would be a private enough place.
Our library has some really creepy stacks (and a sketchy elevator complete with metal accordion door) that are pretty much empty at certain times during the semester.
Of course I’ve had calls where I wanted privacy. But…I’ve never taken it to the point where I had to be 100% sure that there was no possible way I could be overheard. Walking around the main quad is a good way to have a private phone call. I’m walking past people that I do not know, who do not care about the 2 second snippet of my conversation that they “overheard” as I was walking past them.
Oh, so it’s the fact that OP wants 100% privacy that people are getting hung up on? Haha…ok. 
Back in the day before cell phones, I made my personal phone calls when I knew my roomate would be out of the room, It was the only way to ensure privacy. Today, you can walk anywhere you want as you are chatting. I would look for the most private, empty place to make a call that had low traffic flow, and walk away if need be to maintain privacy.
Our library has a little tardis shaped room that’s soundproof. It’s designed for phone calls. (Or so they claim.)
Now I’m interested as to what the phone call is about. I have some ribald suggestions, but maybe I should refrain…