But medical checks have revealed only a partial groin tendon tear, which means Jones will not have to undergo surgery.
Had the 29-year-old required an operation, then he would have effectively followed injured Wales colleagues Gavin Henson and Ryan Jones on to the World Cup scrapheap.
Jones will miss Wales' tournament warm-up games against Argentina and France in Cardiff later this month, but he is expected to regain full fitness shortly afterwards.
Wales' opening World Cup pool game is against Canada in Nantes on September 9.
Jenkins said: "This is magnificent news for us, and just the tonic we need as we prepare for our warm-up games against England this week and Argentina and France at the Millennium Stadium later this month.
"Stephen is very much the lifeblood of this squad.
"He is a key leader and vital to us, not just because of his rugby experience and what he offers on the pitch, but because of the impact he makes on the squad throughout our time together.
"He displays the sort of attitude that is reflective of the positive, enthusiastic and professional mood in the squad we have assembled."
Jones captained Wales during last season's RBS 6 Nations Championship, and although he is under acute pressure for the number 10 shirt from Ospreys prodigy James Hook, his experience will be crucial in France.