I don't know if this is THE version of the story, but here's A version of the story.
http://www.christendom.edu/life/stre...en_ribbon.html
The title I searched for was "The Green Ribbon" which is what it was called when I read it originally many long years ago.
EDIT: I found a book that has it with a few other classic scary stories. It's called "In a Dark, Dark Room and Other Scary Stories" by Alvin Schwartz. Here's the Amazon link:
http://www.amazon.com/Dark-Room-Othe.../dp/0064440907
If it is in one of the "Scary Stories to Tell in the Dark" books, it has been renamed to something not obviously about ribbons or strings so it might be there, but I can't figure out which one (if at all) just based on the story titles from Wikipedia.