They've begun their discussion in order to appoint a new bishop for the diocese.
It's after an electoral college made up of 47 people including bishops, clergy and laity, could not come to a decision last month on who should succeed former Bishop of Llandaff and Archbishop of Wales, Dr Barry Morgan.
They needed a two-thirds majority in order to pick a successor but couldn't reach it. It's been reported that there was a difference of opinion over one of the candidates, who is openly gay.
Dr Barry Morgan, who retired earlier this year, had previously declared his support for gay marriage - something not currently legal in the Church in Wales.
Church in Wales rules state that because the electoral college could not come to a decision on a new bishop, the decision is now taken out of their hands and given solely to the five remaining diocese bishops in Wales to make the choice themselves.
They're picking from a selection of clergy put forward in a consultation that was held after the electoral college failed to pick a new bishop.
A decision to select a new Archbishop of Wales then goes back to the electoral college.
