Revd Pat (Patricia) Storey succeeds church of Ireland Primate, Archbishop Richard Clarke of Armagh, in her new role.
Responding to the news, Archbishop Clarke said he was “certain that her ministry in the Dioceses of Meath and Kildare and the wider Church will be a blessing to many.”
The 53-year-old is married with two adult children and is currently Rector of St Augustine’s Parish in Derry since 2004.
"I have had an extraordinarily happy experience in St Augustine's and in this wonderful city, which I will be sad to leave," she said.
"However, I count it an enormous privilege to begin a new phase of my ministry with the people of Meath and Kildare, and I look forward to working with the team of clergy who are already there.
"I would sincerely ask for your prayers for myself and my family, who are the best family in the world."
The historic appointment is a new departure for the Church of Ireland.
Bishop-elect Story grew up in Belfast and studied French and English at Trinity College, Dublin.
She was ordained a priest in 1998 after her training at the Church of Ireland Theological College.
The appointment was welcomed by Watch, a group that campaigns for the appointment of women bishops.
"We're delighted and thrilled and we warmly congratulate her," the Rev Lindsay Southern, a spokeswoman for the group told the BBC.