The Wales players will begin preparations for next Saturday's Millennium Stadium rematch against England without the popular Scarlets back.
Stoddart, 26, broke his left leg during Wales' 23-19 defeat at Twickenham on Saturday and is expected to face a lengthy spell out of the game.
His absence will be a major blow for Wales coach Warren Gatland, who was unable to pick injured full-back Lee Byrne (knee) and wing Leigh Halfpenny (ankle) against England.
Stoddart scored two tries from three starts during last season's RBS 6 Nations Championship, and he was expected to feature among the 30-man World Cup squad Gatland will announce two weeks' today.
Stoddart suffered the injury when he was tackled by England full-back Delon Armitage, having just intercepted a pass, and the extent of it quickly became clear.
Wales centre Jonathan Davies, a Scarlets colleague of Stoddart's, added: "It is terrible news. I am gutted for him because he has worked so hard to get here in the last couple of months.
"I know rugby is a hard game, but when you see a friend that you play rugby with every day pick up an injury like that, it is tough to take.
"I just wish him a speedy recovery and hope he gets back to full fitness."
Gatland added: "You have got to feel for him.
"We have been joking in the past, calling him 'glass' and 'crystal' because he does pick up a few injuries, but to get an injury like that - you hope he recovers as quickly as possible.
"Morgan looked dangerous with the ball in hand, he looked threatening. It was a shame he picked up that injury."
Sympathy and concern was etched all over the England players' faces after Stoddart went down near the touchline in front of Twickenham's East Stand.
"Jamie Roberts was on the floor saying he (Stoddart) was in trouble," recalled England scrum-half Danny Care, who helped stop the game so Stoddart could be treated.
"I think he was screaming. He was definitely in pain, so there it was safety first, stop play and get him off the pitch.
"If someone is injured you want to stay away from him, you don't want to fall on him or anything. Anyone would have stopped the play.
"It is such a sad way to get injured. That is the sport we are in. It is a tough, physical sport and injuries are going to happen. You have to be lucky over the next few games (to make the World Cup)."
Gatland is expected to make a number of changes for England's Cardiff visit, with centre Gavin Henson, prop Gethin Jenkins and number eight Gareth Delve among those waiting to discover if they will be involved.
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