Her appointment as the new Bishop of Meath and Kildare was announced by the Church on Friday following a meeting of the House of Bishops.
She succeeds the Most Reverend Dr Richard Clarke, who is now Archbishop of Armagh.
Her appointment by the House of Bishops this week came after the Episcopal Electoral College failed to elect Dr Clarke's successor during their meeting on 28 May.
Archbishop Clarke said: "Having known Pat Storey since she was an undergraduate and I was Chaplain at Trinity College, Dublin, I very much welcome her as a new bishop.
"She is a person of great warmth, intelligence and spiritual depth and I am certain that her ministry in the Dioceses of Meath and Kildare and the wider Church will be a blessing to many. We remember her and her family in our prayers."
Mrs Storey, 53, is currently Rector of St Augustine's Parish Church, Londonderry, a position she has held since 2004.
After studying French and English at Trinity College, Dublin, she trained at the Church of Ireland Theological College - now Institute - and was ordained deacon in 1997 and priest in 1998.
She is a member of the Standing Committee of the General Synod.
Commenting on her appointment as Bishop of Meath and Kildare, she said: "I am both excited and daunted by this new adventure in our lives. I have had an extraordinarily happy experience in St Augustine's and in this wonderful city which I will be sad to leave.
"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."
Details of her consecration and enthronement will be announced at a later date.
The Archbishop of Dublin, the Most Reverend Dr Michael Jackson said her election would bring "delight" to many Anglicans.
"Pat herself brings to this work of God a warm personality and a breadth of spiritual gifts to share generously in the church and in the community," he said.
"I look forward to having Pat as a colleague and neighbour and I encourage everyone to pray for her, for Earl and for their family as she embarks on this fresh journey of service and leadership of God's people."
Bishop of Derry and Raphoe, the Right Reverend Ken Good, said her appointment was "great news".
"In the Dioceses of Derry and Raphoe we know well how gifted the Reverend Pat Storey is and how effective her ministry is not only in a parish setting but also far beyond the local church," he said.
"Her warmth, her deep personal faith and her natural ability to relate to people have enabled her to make a significant impact throughout the city, the diocese and the wider community.
"I am convinced that Pat will be an excellent bishop. Not only will the Dioceses of Meath and Kildare benefit greatly from her episcopal ministry but so, too, will the wider church and community.
"The parish of St Augustine's and the Dioceses of Derry and Raphoe will greatly miss Pat and her husband, the Revd Earl Storey. Yet we are really delighted with this news and assure them of our prayers for all that lies ahead."