NHS Lothian has admitted hand washing and cleanliness failures are likely to be responsible for the transmission of the virus, which is usually spread by contact with blood.
It is understood the hospital became aware of the problem when the patient, who was treated at the emergency department of Edinburgh Royal Infirmary (ERI) in July, became unwell with symptoms of hepatitis C several weeks later.
It emerged the patient had been in the A&E at the same time as a hepatitis C sufferer. An in-depth investigation identified the genetic fingerprints of the viruses and both patients had the same strain.
Experts said patient-to-patient spread of hepatitis C had only ever happened on a "handful" of occasions in Scotland.
Eminent microbiologist Professor Hugh Pennington described the situation as a "major incident". He said: "It is something entirely preventable because we know exactly how the virus is transmitted and we know we can prevent it."
NHS Lothian has identified 34 other patients who were also looked after in the department at the same time and offered them information, advice and testing as a precaution. A dedicated NHS 24 helpline has also been set up to answer questions.
Dr David Farquharson, medical director of NHS Lothian, said: "We take infection prevention and control extremely seriously and the safety of our patients is our top priority.
"Unfortunately on this occasion we have fallen below the high standards which we expect. We have spoken to and apologised to the individual involved, and I would like to take this opportunity to publicly apologise for the distress this has caused.
"This investigation has identified failings in our processes and these are being urgently addressed."
He stressed the risk of anyone else being infected was very low.
Hepatitis C is a blood borne virus that can damage the liver. It is largely associated with needle sharing among drug users and more than 90% of the 38,000 people infected with chronic hepatitis C in Scotland became ill this way.
Patients have also contracted the virus after receiving contam-inated blood through transfusions and haemophilia treatments. Hospitals use disposable syringes and since 1991 all blood donations have been screened for the virus.
Asked about how hepatitis C could have spread between patients in the ERI A&E, Dr Farquharson said it was likely to do with a combination of "environmental contamination and inadequate hand hygiene".
He added: "We are reviewing the infection control measures across all our hospitals."
Hepatitis C can be treated with antiviral medicines that stop it from multiplying inside the body.
Professor David Goldberg, consultant epidemiologist for Health Protection Scotland, said: "You are much more likely to respond to therapy if you have got mild disease than if you have got more advanced disease. Even people with more advanced disease can respond to therapy. As a principle it is better to be treated earlier than later."
Neil Findlay, Scottish Labour health spokesman, questioned why NHS Lothian first announced details of the incident yesterday when the problem originated in July.
The NHS 24 helpline can be contacted on 08000 28 28 16.