She will be the eighth female bishop to be announced by the Church of England.
The Bishop of Sherborne works in the Diocese of Salisbury and mainly has responsibility for parishes in Dorset.
Archdeacon Gorham said it had been "a real privilege" to serve the church in Buckinghamshire and work in the Diocese of Oxford. The Right Reverend Graham Kings left the post in July.
She said: "I now look forward to getting to know the people and places of Dorset, an area I have loved since childhood holidays."
The Bishop of Salisbury, the Right Reverend Nicholas Holtam, said Archdeacon Gorham "emerged as the right person for this post from a company of excellent men and women considered equally".
She will be consecrated as a bishop on 24 February at Westminster Abbey.
Women Bishops in the Church of England
The Archdeacon of Buckingham, the Venerable Karen Gorham, is to be the new Bishop of Sherborne.