According to St. Anthony Messenger's columnist Albert Haase, O.F.M., in the January 1999 issue, St. Francis had nothing to do with writing the prayer. The earliest version has been found in the breviary of England's William the Conqueror, king from 1066 to 1087. That's nearly 200 years before Francis of Assisi.
According to Haase, Cardinal Francis Spellman attached the name of St. Francis, his patron saint, to the prayer. In visiting Assisi to celebrate his appointment to the College of Cardinals, he found the prayer under the title of "A Simple Prayer" with a picture of St. Francis.
After the cardinal returned to the United States, he passed out copies under the title of "The Peace Prayer of St. Francis."
While Francis did not write the prayer, it is very much in his spirit.