Yes, there are print-on-demand publishers who will do this no problem. Here in the U.S., I know of CreateSpace and Lulu. If you live somewhere else, try doing a google/bing/whatever search for "print on demand publishers". Check out their terms. You want someone who doesn't charge any non-trivial one-time fee to get started. (Like in the U.S., Lightning Source is the other big POD publisher, but they'd be a poor choice for one copy as they charge a one-time setup fee, I think it was $75 last time I checked but that was years ago.)
I do this every year. I like a calendar/appointment book in a certain format. I used to search office supply stores every year to find something resembling what I want. Now I just make my own. CreateSpace charges me something like $7 to print and ship it. I've done this for, I don't know, 5 or 6 years now, and they've never complained about me printing just 1 or 2 copies.