The group responsible for appointing a new bishop couldn't come to a two-thirds majority on who Dr Morgan's successor should be as Bishop of Llandaff after he retired from that post, and also as Archbishop of Wales, on his 70th birthday at the end of January.
The Church in Wales has not said why the Electoral College - made up of bishops, clergy and laity - was unable to agree on the next Bishop of Llandaff during a three-day, closed-door discussion at Llandaff Cathedral (above) in February.
Some have suggested there was disagreement over appointing Jeffrey John - an openly gay clergyman.
Now a consultation process has been launched where the names of potential candidates will be produced from the various Welsh dioceses before a second three-day lock-in discussion at Llandaff Cathedral in March, in the hope of agreeing on a successor for the diocese.
Once a new a bishop is selected, a process will begin to choose a new Archbishop of Wales from the group of six Welsh bishops.
The Church in Wales said: "The Bishops continue to ask for the prayers of the church both for the Diocese of Llandaff and for their own work as they continue to discharge their responsibility for discerning the person whom they believe will serve not only the Diocese of Llandaff but also the wider church in the office of Bishop."
