Morgan damaged knee ligaments, and will have to undergo an operation to correct the damage.
Medical reports suggest he will not be back in action before September at the earliest, making his participation in the forthcoming World Cup seem unlikely.
The injury is particularly cruel for Morgan, who missed the last World Cup with a similar knee problem.
Wales coach Steve Hansen said.
"It is really bad luck for him because he's a top lad.
"We wish him well and I'm sure all Welsh supporters wish him well too."
"Kevin will probably need an operation and he is unlikely to be back before September."
The Swansea website suggests that Swansea are likely to call on Gareth Swales, Shaun Payne or Gavin Henson to fill the full-back berth for the rest of the campaign with Morgan's enforced absence.
All Whites coach John Connolly echoed the disappointment of the player.
"It's dreadful luck for Kevin, we all hope he makes a quick recovery from the operation and he can return in time for the World Cup."