It is the latest blow to hit Mike Ruddock's squad ahead of their championship defence in the New Year and Morgan will join Ryan Jones and Brent Cockbain among the injured players who will sit out the RBS Six Nations campaign.
Also missing for the first three rounds of matches will be Gavin Henson, who was suspended for ten weeks and two days following his citing for using an elbow in the Heineken Cup tie between the Ospreys and Leicester Tigers before Christmas.
Tuesday's defeat at the Arms Park provided another sickening injury blow for the Dragons, who meet the Ospreys on Sunday and Connacht on 5th January, with a broken thumb for international wing Nathan Brew. In only his second game back after nine months on the sidelines, Brew sustained the same injury that centre Hal Luscombe endured and will also undergo surgery this week which will keep him out of action for two months.
"Kevin will have an operation next week and will be out for four months which effectively rules him out for the rest of the season," said Dragons Rugby Manager Jim McCreedy. "It is a bitter blow for Morgan as he is an influential player both for The Dragons and Wales, we wish him a speedy recovery.
"Nathan will be out for two months and it is a bitter blow for him, especially as he has just returned from injury and was showing signs of his form when he was capped by Wales. He will not be available to us until early March and again we wish him a speedy recovery."