Cardiff Blues flanker Martyn Williams will replace him as skipper, with Neath-Swansea Ospreys forward Jonathan Thomas wearing the number six jersey alongside Williams and Dafydd Jones in the back-row.
There are three other changes, all up front, from the Scotland clash.
Llanelli Scarlets prop Iestyn Thomas starts instead of Duncan Jones, with fit-again World Cup forwards Robin McBryde and Robert Sidoli also returning to the starting XV.
Scarlets hooker McBryde takes over from Mefin Davies, who drops to the bench, while Celtic Warriors lock Sidoli ousts Gareth Llewellyn.
Wales coach Steve Hansen though, has retained the back division performed so well at the Millennium Stadium.
"It is disappointing to lose Colin, who has been playing good Test rugby as well as growing into a strong leader," said Hansen.
"We have taken the decision which allows Colin's injury to be rectified once and for all, making him available for the rest of the tournament.
"We are very lucky that we have able replacements in Jonathan Thomas and Alix Popham (on the bench) from a playing point of view, and Martyn Williams from a leadership perspective.
"It's now important that our designated lieutenants stand up and be counted in helping Martyn lead the side. We are looking forward to travelling to Dublin, and playing a game of rugby we can all be proud of."
Hansen added that prop Jones had been omitted for "tactical reasons," and stressed that it was not a form issue.
Wales team to play Ireland in RBS 6 Nations match in Dublin on Sunday:
G Thomas (Celtic Warriors); R Williams (Cardiff), Parker (Celtic Warriors), Harris (Cardiff), S Williams (Neath-Swansea); S Jones (Llanelli), Cooper (Celtic Warriors); I Thomas (Llanelli), McBryde (Llanelli), A Jones (Neath-Swansea), Cockbain (Celtic Warriors), Sidoli (Celtic Warriors), J Thomas (Neath-Swansea), M Williams (Cardiff, capt), D Jones (Llanelli).
Reps: Davies (Celtic Warriors), Jenkins (Celtic Warriors), Michael Owen (Gwent), Popham (Leeds), Peel (Llanelli), Sweeney (Celtic Warriors), Shanklin (Cardiff).