Known as Revd Libby Lane, she becomes the first female bishop in the Church of England.
Speaking about the appointment she said: "This is unexpected and very exciting.
"On this historic day as the Church of England announces the first woman nominated to be Bishop, I am very conscious of all those who have gone before me, women and men, who for decades have looked forward to this moment.
"But most of all I am thankful to God."
Revd Libby Lane is married to George, a chaplain at Manchester Airport and has two grown up children.
Her interests include being a school governor, encouraging social action initiatives, learning to play the saxophone, supporting Manchester United, reading and doing cryptic crosswords.
Her appointment comes just months weeks after Parliament pushed through legislation to allow women to take on the Church's most senior roles.
Speaking about the appointment Prime Minister David Cameron said: "I would like to congratulate Elizabeth Lane on her appointment as Suffragan See of Stockport and in doing so becoming the first woman bishop in the Church.
"This is an historic appointment and an important step forward for the Church towards greater equality in its senior positions.