Archbishop Kevin McDonald, 62, has had a triple heart bypass.
He also suffers from severe osteoarthritis which has already necessitated surgery and he may need more next year.
Archbishop McDonald, who was installed in Southwark six years ago, said: "I feel great sadness at having to relinquish my post as Archbishop of Southwark."
He added: "Although I have had to contend with illness over the last three years, this appointment has been a great grace.
"It has been a privilege to lead this great diocese and I have received a wonderful response to everything I have tried to do.
"I have also been very appreciative of the prayers of so many people while I have been ill.
"The diocese will continue to be very much in my thoughts and prayers in the time ahead."
Archbishop McDonald, who became a priest in 1974, was born in Stoke-on-Trent, Staffordshire.
A spokesman for the Catholic Church in England and Wales said Pope Benedict XVI had accepted Archbishop McDonald's resignation on Friday.