The baby boy, who was believed to be no more than an hour old when he was left at the Emergency Department, is said to be doing well.
However, despite a high-profile appeal, the mother has not come forward.
The investigation was launched amid concerns for the safety and welfare of the mother, and special hotlines were set up for the woman, her family or friends to contact Health Department staff.
A police spokesperson said: ‘The police investigation launched to find the mother of the baby left at the hospital a month ago received a positive response from the public, who clearly wanted to help.
‘Officers have carried out extensive enquiries and explored numerous leads. However, all these leads have now been exhausted without conclusion. Consequently the information that has been received will remain on file in case any new information comes to light.
‘The baby is being cared for and doing well. However, in order to protect his anonymity there will be no further comment on his ongoing care.’