The 27-year-old, who has played 48 times for Wales, suffered the injury for his club side Clermont Auvergne against Montpellier.
A statement on the WRU website ( said he was likely to be out for between six and seven months.
The World Cup, being played in England and Wales, starts in September. Wales are in Pool A with Australia, England, Fiji and Uruguay.
Wales head coach Warren Gatland said: "It's a disappointment for Jonathan and for us, but the advice is for him to have the surgery straight away.
"Obviously, the injury rules him out of the 2015 Rugby World Cup but our focus is his long-term career and we will be working closely with him during his recovery looking towards the 2016 Six Nations."

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