Both doctors travelling with the England team assessed the severity of the problem and came to the same conclusion; the Manchester United man would play no part in the competition.
"Further to a scan arranged by the England medical staff immediately after training on Friday afternoon, Rio has been ruled out for 4-6 weeks and will therefore unfortunately miss the World Cup," said a statement.
"Tottenham Hotspur's Michael Dawson has been contacted and will fly to South Africa.
"Steven Gerrard will lead the England team as captain in the tournament."
It is a grievous blow for coach Fabio Capello, who rated Ferdinand as one of the most influential players in his squad as well as a lynchpin in his defence.
He must now decide who to replace Ferdinand with, having added Michael Dawson to a list of central defenders that also includes Ledley King, Matthew Upson and Jamie Carragher.
"It is obviously bad news and everyone with the squad is very disappointed and sorry for Rio," said Capello.
"It was an accidental injury in training, but had nothing to do with the pitch."