He has undergone successful surgery and is now in a comfortable condition and will remain in hospital during his recovery.
Any updates will be given at the appropriate time and in conjunction with medical staff and in line with the wishes of his family.
Chief Executive of the Scarlets Mark Davies said: "All our thoughts and best wishes as Scarlets are with Rhys and his wife Paula and their family and the region will be keeping in close contact with medical teams at Morriston Hospital as to his progress but it is understood that he is comfortable and doing well.
"Out of respect for Rhys and at the request of his family, we would ask that this matter is treated with dignified sensitivity and that the family and his colleagues at Parc y Scarlets are given time and space to concentrate on the most important focus which is supporting Rhys' wellbeing and recovery.
"I am sure the whole rugby community in Wales would join with us in expressing to Rhys - all our very best wishes for a quick and comfortable recovery."