The 65-year-old will leave his post as Bishop of Hull before the end of the year after 16 years in his current post.
Bishop Frith said he was delighted with his new post: "I am thrilled to have been chosen to be the next Bishop of Hereford.
"The Diocese has a wonderful Christian heritage and I greatly look forward to playing my part in building up the church and serving the wider community."
The Bishop has worked in a mix of urban and rural areas in his forty years of ministry and trained at St John's College Nottingham.
He has four children, four step-children and is a grandfather of seven.
The Archbishop of York, Dr John Sentamu said: "Hull is a wonderful city, it is so full of energy and life.
Bishop Richard has been a great Ambassador for Urban Life and Faith for Hull - and has brought his passion and joyful enthusiasm to his mission for the Diocese of York.
"His servant-leader ministry over the last 16 years has focused very much on the encouragement of others, on building friendships, and growing discipleship across the whole of the East Riding.
"He goes from Hull to Hereford, with our prayers and our thanks - he is a great friend and a good shepherd. I know that he and Kay will be joyfully welcomed to the Diocese of Hereford."