The match took place on Friday, but the extent of Ansbro's injury was only revealed earlier today.
"Joe broke a small bone at the top of his neck early in the game's second half," read a statement released by London Irish.
"After an MRI scan, it was confirmed that immediate surgery was not required, and the injury would be managed by external fixation."
Scotland team doctor James Robson believes that the injury could keep Ansbro out for almost three months, meaning that he will miss Scotland's autumn internationals against New Zealand, South Africa and Tonga.
"The management of the injury Joe has sustained typically involves a period of between six and eight weeks' immobilisation," added Robson.
"After that there will be a further four to six weeks of rehabilitation before he can look at a return to the game. We wish Joe well in his recovery."
Ansbro has made nine appearances for Scotland, scoring two tries.