Melissa Priestley, 33, worked at HMP Low Newton in Durham, the prison where serial killer Rose West is held.
Priestley will be sentenced next month after appearing at Durham Crown Court to admit misconduct in public office.
An investigation began following a tip-off, before letters to Priestley were found in the prisoner's cell.
Investigators also found messages on Priestley's mobile phone which corroborated the relationship, the CPS said.
John Dilworth, of the CPS North East, said: "For people to have confidence in the Criminal Justice System, they need to know that the law applies equally to all of those involved in the delivery of justice.
"The relationship between Melissa Priestley and the prisoner, over whom she had a professional duty of care, was wholly inappropriate.
"I would like to praise the swift actions of the prison authorities and police, once the initial reports of this relationship were received.
"Through their diligence vital evidence was preserved, assisting greatly in the Crown's preparation of a robust case against Melissa Priestley."
Priestley was bailed pending a sentencing hearing on a date yet to be set.
A prison officer at a women's jail has admitted having an inappropriate relationship with an inmate, the Crown Prosecution Service (CPS) has said.