The centre suffered the injury in Friday's pre-season clash with Munster - with Ansbro having remained in University Hospital, Cork, to receive treatment.
However the initial assessment has been promising and the 26-year-old could be back playing in three months.
A statement from London Irish read: "Joe broke a small bone at the top of his neck early in the game's second half. After An MRI scan, it was confirmed that immediate surgery was not required, and the injury would be managed by external fixation.
"He is in the care of the club's and hospitals medical team and is expected to return to the UK later this week."
Scotland have remained in contact with the Exiles and they are confident Ansbro will be back playing soon.
Scotland team doctor James Robson said: "The management of the injury Joe has sustained typically involves a period of between six-eight weeks immobilisation then a further four-six weeks of rehabilitation before he can look at a return to the game. We wish Joe well in his recovery."
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