NHS Lothian is trying to find the source of the infection but believes it is likely to have come from cross-contamination in the A&E department of Edinburgh Royal Infirmary in July.
A total of 34 other patients who were in the emergency ward at the same time have been contacted by the health board. Their risk of infection is thought to be low but they are being offered precautionary testing, as well as advice and information.
Infection-control measures for the emergency department have been reviewed, and extra measures have been put in place, according to NHS Lothian.
The health board also says it has stressed the importance of hand hygiene and the risks associated with blood-borne viruses, such as hepatitis C, to its staff.
A helpline has been set up for any members of the public who have questions about the disease, on 08000 282 816.
Around 1% of the population of Scotland is infected with hepatitis C and while some people experience little liver damage, some with the disease can develop cirrhosis of the liver, liver cancer or end-stage liver disease.
Hepatitis C can be treated with anti-viral medicine that stops it from multiplying within the body and help prevent liver damage.
Dr David Farquharson, medical director at 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.
"I would like to reassure everyone who has visited our emergency department that the risk of being infected is very low. If anyone is at all concerned, I would encourage them to contact the dedicated helpline for more information and advice."